<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lesson17intro" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">Introduction to the Uncertainty Principle</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson17_intro">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson17_intro" data-block-type="html" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
Now, we delve further into the relation between waves in coordinate space vs. frequency space. We have seen that, through a Fourier transform, a temporal signal [mathjaxinline]f(t)[/mathjaxinline] can be expressed in terms of the amplitudes of its frequency components, represented by [mathjaxinline]c(\omega )[/mathjaxinline]. Now, we show that there are deeper relations between a function and its Fourier conjugate (e.g., between [mathjaxinline]f(t)[/mathjaxinline] and [mathjaxinline]c(\omega )[/mathjaxinline]). </p><p>
One of the interesting ties between Fourier conjugate functions is the <i class="itshape">uncertainty principle</i>—when a function like [mathjaxinline]f(t)[/mathjaxinline] exhibits less "spread," its Fourier conjugate [mathjaxinline]c(\omega )[/mathjaxinline] exhibits more "spread." </p><p>
Of course, we will formalize what we mean by "spread." In doing so, we derive a mathematically rigorous and generalizable principle, which plays a familiar role in quantum mechanics, but whose origin is fundamentally mathematical. Ultimately, the <i class="itshape">uncertainty principle</i> is relevant to myriad systems, from classical waves to quantum waves. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L26v1" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v1: Review of Fourier transform properties</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v1" data-block-type="video" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v1: Review of Fourier transform properties</h3>
<div
id="video_L26v1"
class="video closed"
data-metadata='{"prioritizeHls": false, "sources": [], "autoplay": false, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v1/handler/transcript/translation/__lang__", "speed": null, "generalSpeed": 1.0, "streams": "1.00:xSURMPMS_cU", "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "completionPercentage": 0.95, "start": 0.0, "savedVideoPosition": 0.0, "autohideHtml5": false, "saveStateEnabled": false, "recordedYoutubeIsAvailable": true, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v1/handler/publish_completion", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v1/handler/xmodule_handler/save_user_state", "poster": null, "duration": 0.0, "captionDataDir": null, "showCaptions": "true", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v1/handler/transcript/available_translations", "ytTestTimeout": 1500, "transcriptLanguages": {"en": "English"}, "completionEnabled": false, "ytMetadataEndpoint": "", "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L26v1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L26v1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/3ccf32f9906d593a751a2c7cbd40b1ed/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L26v1.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_16_ex00" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26Q1: Fourier Transform Example</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_00">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_00" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_16_00" class="problems-wrapper" role="group"
aria-labelledby="lect_16_00-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_00" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_00/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_16_00-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_00-problem-progress" tabindex="-1">
Fourier Transform Example
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_00-problem-progress"></div>
<div class="problem">
<div>
<p>
Calculate the Fourier transform, [mathjaxinline]c(\omega )[/mathjaxinline], of the following function: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]f(t)=\cos (\omega _{0}t)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Choose the correct answer choice below. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_16_00_2_1">
<fieldset aria-describedby="status_lect_16_00_2_1">
<div class="field">
<input type="radio" name="input_lect_16_00_2_1" id="input_lect_16_00_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_16_00_2_1-choice_1-label" for="input_lect_16_00_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_16_00_2_1">
<text> a) [mathjaxinline]c(\omega )= \frac{\sin (\omega _{0}t)}{\omega _{0}}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_16_00_2_1" id="input_lect_16_00_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_16_00_2_1-choice_2-label" for="input_lect_16_00_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_16_00_2_1">
<text> b) [mathjaxinline]c(\omega )= \delta (\omega )[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_16_00_2_1" id="input_lect_16_00_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_16_00_2_1-choice_3-label" for="input_lect_16_00_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_16_00_2_1">
<text> c) [mathjaxinline]c(\omega )= \frac{1}{2} \left[\delta (\omega +\omega _{0}) - \delta (\omega -\omega _{0})\right][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_16_00_2_1" id="input_lect_16_00_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_16_00_2_1-choice_4-label" for="input_lect_16_00_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_16_00_2_1">
<text> d) [mathjaxinline]c(\omega )= \frac{1}{2} \left[\delta (\omega +\omega _{0}) + \delta (\omega -\omega _{0})\right][/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_16_00_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_16_00_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_16_00_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Fourier Transform Example" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_16_00" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_16_00">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_00-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_00-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_00-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_16_ex01" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26Q2: Comparing f(t) and c(w) [SIMULATION]</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_01">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_01" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_16_01" class="problems-wrapper" role="group"
aria-labelledby="lect_16_01-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_01" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_01/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_16_01-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_01-problem-progress" tabindex="-1">
Visualizing f(t) and c(w)
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_01-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider examples of Fourier transform pairs, [mathjaxinline]f(t)[/mathjaxinline] and [mathjaxinline]c(\omega )[/mathjaxinline], shown in the figures below. The parameter that determines the "spread" of [mathjaxinline]f(t)[/mathjaxinline] is different in each plot. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/8f556a413879769f4f69627757599634/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_16_01a.png" width="660"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/33047528415edd91103a686d328507b8/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_16_01b.png" width="660"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/f361ef7403469b16e2c3705bc952642c/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_16_01c.png" width="660"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
Select ALL true statements that apply, in general, to Fourier pairs. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_16_01_2_1">
<fieldset aria-describedby="status_lect_16_01_2_1">
<div class="field">
<input type="checkbox" name="input_lect_16_01_2_1[]" id="input_lect_16_01_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_16_01_2_1-choice_0-label" for="input_lect_16_01_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_16_01_2_1">
<text>a) The "spread" of [mathjaxinline]f(t)[/mathjaxinline] and [mathjaxinline]c(\omega )[/mathjaxinline] are always inversely related</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_16_01_2_1[]" id="input_lect_16_01_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_16_01_2_1-choice_1-label" for="input_lect_16_01_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_16_01_2_1">
<text>b) The shape of [mathjaxinline]c(\omega )[/mathjaxinline] will always be a Gaussian function</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_16_01_2_1[]" id="input_lect_16_01_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_16_01_2_1-choice_2-label" for="input_lect_16_01_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_16_01_2_1">
<text>c) There exists a function [mathjaxinline]f(t)[/mathjaxinline] whose Fourier transform [mathjaxinline]c(\omega )[/mathjaxinline] has the same functional form as [mathjaxinline]f(t)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_16_01_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_16_01_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_16_01_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Visualizing f(t) and c(w)" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_16_01" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_16_01">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_01-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_01-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_01-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@SIM_lect_16_01">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@SIM_lect_16_01" data-block-type="html" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=fourier_pair_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/>   [mathjaxinline]\bullet[/mathjaxinline]  git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/>   [mathjaxinline]\bullet[/mathjaxinline]  information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/>   [mathjaxinline]\bullet[/mathjaxinline]  information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/>   [mathjaxinline]\bullet[/mathjaxinline]  information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  scipy
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  numpy
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  ipywidgets
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  nbinteract
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  matplotlib
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  pandas
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  IPython
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  ffmpeg
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/532bc862790161d15700a41243026083/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_fourier_pair_1.png"/></div><p/>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L26v2" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v2: Define the Spread of Time and Frequency</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v2">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v2" data-block-type="video" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v2: Define the Spread of Time and Frequency</h3>
<div
id="video_L26v2"
class="video closed"
data-metadata='{"prioritizeHls": false, "sources": [], "autoplay": false, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v2/handler/transcript/translation/__lang__", "speed": null, "generalSpeed": 1.0, "streams": "1.00:prwb5tLUGEE", "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "completionPercentage": 0.95, "start": 0.0, "savedVideoPosition": 0.0, "autohideHtml5": false, "saveStateEnabled": false, "recordedYoutubeIsAvailable": true, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v2/handler/publish_completion", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v2/handler/xmodule_handler/save_user_state", "poster": null, "duration": 0.0, "captionDataDir": null, "showCaptions": "true", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v2/handler/transcript/available_translations", "ytTestTimeout": 1500, "transcriptLanguages": {"en": "English"}, "completionEnabled": false, "ytMetadataEndpoint": "", "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L26v2"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L26v2">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v2">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/ada5371c29f17f6a377deb463f78906f/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L26v2.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_16_ex02" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26Q3: Calculation of Average Values</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02a" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_16_02a" class="problems-wrapper" role="group"
aria-labelledby="lect_16_02a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_16_02a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02a-problem-progress" tabindex="-1">
Calculation of Average Values - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02a-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider the function introduced earlier in this lesson: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]f(t)=e^{-\Gamma |t|}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Calculate the mean value of [mathjaxinline]t[/mathjaxinline]. Express your answer in terms of <code>Gamma</code> for [mathjaxinline]\Gamma[/mathjaxinline], <code>t</code>, and other relevant constants if necessary. </p>
<p>
<p style="display:inline">[mathjaxinline]\langle t \rangle =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_16_02a_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_16_02a_2_1" id="input_lect_16_02a_2_1" aria-describedby="status_lect_16_02a_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_16_02a_2_1"/>
<span class="status unanswered" id="status_lect_16_02a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_16_02a_2_1" class="answer"/>
<div id="display_lect_16_02a_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_16_02a_2_1_dynamath" name="input_lect_16_02a_2_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_16_02a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Calculation of Average Values - part a" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_16_02a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_16_02a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_02a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_02a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_02a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02b" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_16_02b" class="problems-wrapper" role="group"
aria-labelledby="lect_16_02b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_16_02b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02b-problem-progress" tabindex="-1">
Calculation of Average Values - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_02b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> Calculate the mean value of [mathjaxinline]t^{2}[/mathjaxinline]. Express your answer in terms of <code>Gamma</code> for [mathjaxinline]\Gamma[/mathjaxinline], <code>t</code>, and other relevant constants if necessary. </p>
<p>
<p style="display:inline">[mathjaxinline]\langle t^2 \rangle =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_16_02b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_16_02b_2_1" id="input_lect_16_02b_2_1" aria-describedby="status_lect_16_02b_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_16_02b_2_1"/>
<span class="status unanswered" id="status_lect_16_02b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_16_02b_2_1" class="answer"/>
<div id="display_lect_16_02b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_16_02b_2_1_dynamath" name="input_lect_16_02b_2_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_16_02b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Calculation of Average Values - part b" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_16_02b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_16_02b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_02b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_02b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_02b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L26v3" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v3: Calculation of the Spread of Frequency</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v3">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v3" data-block-type="video" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v3: Calculation of the Spread of Frequency</h3>
<div
id="video_L26v3"
class="video closed"
data-metadata='{"prioritizeHls": false, "sources": [], "autoplay": false, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v3/handler/transcript/translation/__lang__", "speed": null, "generalSpeed": 1.0, "streams": "1.00:86ShAnOBONY", "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "completionPercentage": 0.95, "start": 0.0, "savedVideoPosition": 0.0, "autohideHtml5": false, "saveStateEnabled": false, "recordedYoutubeIsAvailable": true, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v3/handler/publish_completion", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v3/handler/xmodule_handler/save_user_state", "poster": null, "duration": 0.0, "captionDataDir": null, "showCaptions": "true", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v3/handler/transcript/available_translations", "ytTestTimeout": 1500, "transcriptLanguages": {"en": "English"}, "completionEnabled": false, "ytMetadataEndpoint": "", "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L26v3"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L26v3">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v3">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/8fc8336b6a99d567c89cbdbc97151bac/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L26v3.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_16_checkpoint_1" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">Derivation Checkpoint I</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@review_L26_1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@review_L26_1" data-block-type="html" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><h2>OBJECTIVE</h2></p><p>
Our objective is to compute the following, for the known function [mathjaxinline]c(\omega )[/mathjaxinline]: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax](\Delta \omega )^{2} = \left\langle \left[\omega - \langle \omega \rangle \right]^{2}\right\rangle[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
and write our expression solely in terms of [mathjaxinline]f(t)[/mathjaxinline]. This will enable us to meaningfully compare with [mathjaxinline](\Delta t)^{2}[/mathjaxinline]. </p><p><h2>DERIVATION PART I</h2></p><p>
Here, we will do this derivation a little more completely than done in the lesson video (also fixing some mistakes). Our first objective is to express [mathjaxinline]\langle \omega \rangle[/mathjaxinline] in terms of [mathjaxinline]f(t)[/mathjaxinline]. The definition of [mathjaxinline]\langle \omega \rangle[/mathjaxinline] is: </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\langle \omega \rangle = \dfrac {\int _{-\infty }^{\infty }d\omega \, \omega \left|c(\omega )\right|^{2}}{\int _{-\infty }^{\infty }d\omega \, \left|c(\omega )\right|^{2}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
Define the following variable: </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\eta = \int _{-\infty }^{\infty }d\omega \, \left|c(\omega )\right|^{2} = \frac{1}{2\pi }\int _{-\infty }^{\infty }dt\, \left|f(t)\right|^{2}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p>
where we have used the Fourier defintion </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]f(t) = \int _{-\infty }^{\infty }d\omega \, c(\omega )e^{-i\omega t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table><p>
NOW WE'RE READY TO REALLY START: </p><table id="a0000000006" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000007"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \langle \omega \rangle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\dfrac {\int _{-\infty }^{\infty }d\omega \, \omega \left|c(\omega )\right|^{2}}{\int _{-\infty }^{\infty }d\omega \, \left|c(\omega )\right|^{2}}[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>5</span>)</span></td></tr><tr id="a0000000008"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\dfrac {\int _{-\infty }^{\infty }d\omega \, \omega \left|c(\omega )\right|^{2}}{\eta }[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>6</span>)</span></td></tr><tr id="a0000000009"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \langle \omega \rangle \eta[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int _{-\infty }^{\infty }d\omega \, \omega \left|c(\omega )\right|^{2}[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>7</span>)</span></td></tr><tr id="a0000000010"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int _{-\infty }^{\infty }d\omega \, c^{*}(\omega ) \omega c(\omega )[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>8</span>)</span></td></tr><tr id="a0000000011"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int _{-\infty }^{\infty }d\omega \, \frac{1}{2\pi }\int _{-\infty }^{\infty }dt\, f^{*}(t)e^{i\omega t} \omega \frac{1}{2\pi }\int _{-\infty }^{\infty }dt'\, f(t')e^{-i\omega t'}[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>9</span>)</span></td></tr><tr id="a0000000012"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{4\pi ^{2}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dt'\, dt\, d\omega \, f^{*}(t)\omega e^{i\omega t} f(t')e^{-i\omega t'}[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>10</span>)</span></td></tr><tr id="a0000000013"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{4\pi ^{2}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dt'\, dt\, d\omega \, f^{*}(t)\left(-i\frac{d}{dt} e^{i\omega t}\right) f(t')e^{-i\omega t'}[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>11</span>)</span></td></tr><tr id="a0000000014"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{4\pi ^{2}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dt'\, dt\, f^{*}(t) f(t')\left(-i\frac{d}{dt} \int _{-\infty }^{\infty }d\omega \, e^{i\omega (t-t')}\right)[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>12</span>)</span></td></tr><tr id="a0000000015"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{2\pi }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dt'\, dt\, f^{*}(t) f(t')\left(-i\frac{d}{dt} \delta (t-t')\right)[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>13</span>)</span></td></tr></table><p>
Having used the following identity: </p><table id="a0000000016" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{1}{2\pi }\int _{-\infty }^{\infty }d\omega \, e^{i\omega (t-t')} = \delta (t-t')[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>14</span>)</span></td></tr></table><p>
Continuing, we have: </p><table id="a0000000017" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000018"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \langle \omega \rangle \eta[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{2\pi }\int _{-\infty }^{\infty }dt\, f^{*}(t) \int _{-\infty }^{\infty }dt'\, f(t')\left(-i\frac{d}{dt} \delta (t-t')\right)[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>15</span>)</span></td></tr><tr id="a0000000019"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{2\pi }\int _{-\infty }^{\infty }dt\, f^{*}(t) i\frac{d}{dt} f(t)[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>16</span>)</span></td></tr></table><p>
Having used the following identity: </p><table id="a0000000020" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _{-\infty }^{\infty }dt'\, f(t') \frac{d}{dt} \delta (t-t') = -\frac{d}{dt}f(t)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>17</span>)</span></td></tr></table><p><h2>CONCLUSION PART I</h2></p><p>
We can convert an integral, of the form [mathjaxinline]\int _{-\infty }^{\infty }d\omega \, \omega \left|c(\omega )\right|^{2}[/mathjaxinline] into an integral involving [mathjaxinline]f(t)[/mathjaxinline] and [mathjaxinline]dt[/mathjaxinline], by identifying [mathjaxinline]\omega \rightarrow i\frac{d}{dt}[/mathjaxinline]. </p><p>
With this, we can see what [mathjaxinline](\Delta \omega )^{2}[/mathjaxinline] is: </p><table id="a0000000021" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000022"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle (\Delta \omega )^{2}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \left\langle \left[\omega - \langle \omega \rangle \right]^{2}\right\rangle[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>18</span>)</span></td></tr><tr id="a0000000023"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \dfrac {\int _{-\infty }^{\infty }dt\, \left|\left(i\frac{d}{dt} - \langle \omega \rangle \right) f(t) \right|^{2}}{\int _{-\infty }^{\infty }dt\, \left| f(t)\right|^{2}}[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>19</span>)</span></td></tr></table>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_16_checkpoint_1_prob" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26Q4: Fourier Identity Proof</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@review_L26_1_prob">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@review_L26_1_prob" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_review_L26_1_prob" class="problems-wrapper" role="group"
aria-labelledby="review_L26_1_prob-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@review_L26_1_prob" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@review_L26_1_prob/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="review_L26_1_prob-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@review_L26_1_prob-problem-progress" tabindex="-1">
Fourier Identity Proof
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@review_L26_1_prob-problem-progress"></div>
<div class="problem">
<div>
<p>
In the preceeding review, we stated: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\int _{-\infty }^{\infty }d\omega \, \left|c(\omega )\right|^{2} = \frac{1}{2\pi }\int _{-\infty }^{\infty }dt\, \left|f(t)\right|^{2}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Let's show that this is true. Consider the incomplete steps of the derivation below: </p>
<table id="a0000000003" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000004">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int _{-\infty }^{\infty }d\omega \, \left|c(\omega )\right|^{2}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int _{-\infty }^{\infty }d\omega \, c^{*}(\omega ) c(\omega )[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>1</span>)</span>
</td>
</tr>
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int _{-\infty }^{\infty }d\omega \, \frac{1}{2\pi }\int _{-\infty }^{\infty }dt\, f^{*}(t)e^{i\omega t} \frac{1}{2\pi }\int _{-\infty }^{\infty }dt'\, f(t')e^{-i\omega t'}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>2</span>)</span>
</td>
</tr>
<tr id="a0000000006">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{4\pi ^{2}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dt'\, dt\, d\omega \, f^{*}(t) f(t')e^{i\omega (t-t')}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>3</span>)</span>
</td>
</tr>
<tr id="a0000000007">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \cdots[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>4</span>)</span>
</td>
</tr>
<tr id="a0000000008">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\frac{1}{2\pi }\int _{-\infty }^{\infty }dt\, f^{*}(t) f(t)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>5</span>)</span>
</td>
</tr>
<tr id="a0000000009">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\frac{1}{2\pi }\int _{-\infty }^{\infty }dt\, \left|f(t)\right|^{2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>6</span>)</span>
</td>
</tr>
</table>
<p>
What expression best fills in missing step (4)? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_review_L26_1_prob_2_1">
<fieldset aria-describedby="status_review_L26_1_prob_2_1">
<div class="field">
<input type="radio" name="input_review_L26_1_prob_2_1" id="input_review_L26_1_prob_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="review_L26_1_prob_2_1-choice_1-label" for="input_review_L26_1_prob_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_review_L26_1_prob_2_1">
<text> a) [mathjaxinline]\frac{1}{2\pi }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dt\, d\omega \, f^{*}(t) f(\omega ) \delta (t-\omega )[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_review_L26_1_prob_2_1" id="input_review_L26_1_prob_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="review_L26_1_prob_2_1-choice_2-label" for="input_review_L26_1_prob_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_review_L26_1_prob_2_1">
<text> b) [mathjaxinline]\frac{1}{2\pi }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dt'\, d\omega \, f^{*}(\omega ) f(t') \delta (\omega -t')[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_review_L26_1_prob_2_1" id="input_review_L26_1_prob_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="review_L26_1_prob_2_1-choice_3-label" for="input_review_L26_1_prob_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_review_L26_1_prob_2_1">
<text> c) [mathjaxinline]\frac{1}{2\pi }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dt'\, dt\, f^{*}(t) f(t') \delta (t-t')[/mathjaxinline]</text>
</label>
</div>
<span id="answer_review_L26_1_prob_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_review_L26_1_prob_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_review_L26_1_prob_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Fourier Identity Proof" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_review_L26_1_prob" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_review_L26_1_prob">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="review_L26_1_prob-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="review_L26_1_prob-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="review_L26_1_prob-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L26v4" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v4: Derivation of Uncertainty Principle</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v4">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v4" data-block-type="video" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v4: Derivation of Uncertainty Principle</h3>
<div
id="video_L26v4"
class="video closed"
data-metadata='{"prioritizeHls": false, "sources": [], "autoplay": false, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v4/handler/transcript/translation/__lang__", "speed": null, "generalSpeed": 1.0, "streams": "1.00:3OoDJ9YZqCE", "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "completionPercentage": 0.95, "start": 0.0, "savedVideoPosition": 0.0, "autohideHtml5": false, "saveStateEnabled": false, "recordedYoutubeIsAvailable": true, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v4/handler/publish_completion", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v4/handler/xmodule_handler/save_user_state", "poster": null, "duration": 0.0, "captionDataDir": null, "showCaptions": "true", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v4/handler/transcript/available_translations", "ytTestTimeout": 1500, "transcriptLanguages": {"en": "English"}, "completionEnabled": false, "ytMetadataEndpoint": "", "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L26v4"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L26v4">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v4">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/73dde6d952586acc5b87050012f75649/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L26v4.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_16_checkpoint_2" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">Derivation Checkpoint II</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@review_L26_2">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@review_L26_2" data-block-type="html" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><h2>NEW OBJECTIVE</h2></p><p>
In phase II of this proof, let's recap what we have already shown, which is that [mathjaxinline](\Delta \omega )^{2}[/mathjaxinline] can be expressed in terms of [mathjaxinline]f(t)[/mathjaxinline]: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax](\Delta \omega )^{2} = \dfrac {\int _{-\infty }^{\infty }dt\, \left|\left(i\frac{d}{dt} - \langle \omega \rangle \right) f(t) \right|^{2}}{\int _{-\infty }^{\infty }dt\, \left| f(t)\right|^{2}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
Now our new objective is to compare [mathjaxinline](\Delta \omega )^{2}[/mathjaxinline] and [mathjaxinline](\Delta t)^{2}[/mathjaxinline], since both can be expressed in terms of [mathjaxinline]f(t)[/mathjaxinline]. Recall the following definition of [mathjaxinline](\Delta t)^{2}[/mathjaxinline]: </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax](\Delta t)^{2} = \dfrac {\int _{-\infty }^{\infty }dt\, \left|\left(t - \langle t \rangle \right) f(t) \right|^{2}}{ \int _{-\infty }^{\infty }dt\, \left| f(t)\right|^{2}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p><h2>DERIVATION PART II</h2></p><p>
This part of the proof rests on being a little clever. We will try to sketch out the reasoning in the most transparent way. To start with, if we can come up with a function [mathjaxinline]R(\Delta \omega ,\Delta t)[/mathjaxinline] such that if the function were greater than zero, then we would be able to find an inequality between [mathjaxinline]\Delta \omega[/mathjaxinline] and [mathjaxinline]\Delta t[/mathjaxinline]. One way to write a function greater than zero is: </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]R(\kappa )=\dfrac {\int _{-\infty }^{\infty }dt\, \left|r(\kappa ,t) \right|^{2}}{\int _{-\infty }^{\infty }dt\, \left| f(t)\right|^{2}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p>
Then, [mathjaxinline]R(\kappa ) \geq 0[/mathjaxinline] because both integrands are always positive. Let's define [mathjaxinline]r(\kappa ,t)[/mathjaxinline] to be the following: </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]r(\kappa ,t)= \left(\left[t - \langle t \rangle \right] - i\kappa \left[i\frac{d}{dt} - \langle \omega \rangle \right]\right)f(t)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table><p>
This will be a convenient choice because the integrand [mathjaxinline]|r(\kappa ,t)|^{2}[/mathjaxinline] will generate terms like [mathjaxinline](\Delta \omega )^{2}[/mathjaxinline] and [mathjaxinline](\Delta t)^{2}[/mathjaxinline]. To simplify things, note the following definitions: </p><table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]T=\left[t - \langle t \rangle \right][/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>5</span>)</span></td></tr></table><table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\Omega =\left[i\frac{d}{dt} - \langle \omega \rangle \right][/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>6</span>)</span></td></tr></table><table id="a0000000008" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]r(\kappa ,t)= \left(T - i\kappa \Omega \right)f(t)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>7</span>)</span></td></tr></table><p>
Now let's begin to compute [mathjaxinline]|r(\kappa ,t)|^{2}[/mathjaxinline] (some intermediate steps not shown): </p><table id="a0000000009" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000010"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle |r(\kappa ,t)|^{2}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \left|\left(T - i\kappa \Omega \right)f(t)\right|^{2}[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>8</span>)</span></td></tr><tr id="a0000000011"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \left(T - i\kappa \Omega \right)f(t)\left(T + i\kappa \Omega ^{*}\right)f^{*}(t)[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>9</span>)</span></td></tr><tr id="a0000000012"><td style="width:40%; border:none">Â </td><td style="vertical-align:middle; text-align:right; border:none">
Â
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \left|Tf(t)\right|^{2} + \kappa ^{2}\left|\Omega f(t)\right|^{2} + \kappa T \frac{\partial }{\partial t}\left| f(t)\right|^{2}[/mathjaxinline]
</td><td style="width:40%; border:none">Â </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>10</span>)</span></td></tr></table><p>
Then we can compute each "part" of [mathjaxinline]R(\kappa )[/mathjaxinline] (some intermediate steps not shown): </p><table id="a0000000013" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\dfrac {\int _{-\infty }^{\infty }dt\, \left|T f(t) \right|^{2}}{\int _{-\infty }^{\infty }dt\, \left| f(t)\right|^{2}} = (\Delta t)^{2}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>11</span>)</span></td></tr></table><table id="a0000000014" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\dfrac {\int _{-\infty }^{\infty }dt\, \kappa ^{2}\left| \Omega f(t) \right|^{2}}{\int _{-\infty }^{\infty }dt\, \left| f(t)\right|^{2}} = \kappa ^{2}(\Delta \omega )^{2}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>12</span>)</span></td></tr></table><table id="a0000000015" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\dfrac {\int _{-\infty }^{\infty }dt\, \kappa T \frac{\partial }{\partial t}\left| f(t)\right|^{2}}{\int _{-\infty }^{\infty }dt\, \left| f(t)\right|^{2}} = -\kappa[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>13</span>)</span></td></tr></table><p>
So we are left with the following inequality: </p><table id="a0000000016" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]R(\kappa ,t) = (\Delta t)^{2} + \kappa ^{2}(\Delta \omega )^{2} -\kappa \geq 0[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>14</span>)</span></td></tr></table><p><h2>CONCLUSION PART II</h2></p><p>
The [mathjaxinline]\kappa[/mathjaxinline] value that minimizes [mathjaxinline]R(\kappa ,t)[/mathjaxinline] will yield the strictest inequality. By taking the derivative, it is straightforward to show that this value is [mathjaxinline]\kappa _{\mathrm{min}}=\dfrac {1}{2\Delta \omega ^{2}}[/mathjaxinline], which leads to [mathjaxinline](\Delta t)^{2} - \dfrac {1}{4(\Delta \omega )^{2}} \geq 0[/mathjaxinline], therefore: </p><table id="a0000000017" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\Delta t \Delta \omega \geq \frac{1}{2}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>15</span>)</span></td></tr></table>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_16_ex04a" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26Q5: Review of Derivation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04a" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_16_04a" class="problems-wrapper" role="group"
aria-labelledby="lect_16_04a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_16_04a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04a-problem-progress" tabindex="-1">
Review of derivation
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04a-problem-progress"></div>
<div class="problem">
<div>
<p>
Which of the following relations between [mathjaxinline]f(t)[/mathjaxinline] and [mathjaxinline]c(\omega )[/mathjaxinline] MUST be true in order for the uncertainty relation [mathjaxinline]\Delta t \Delta \omega \geq \frac{1}{2}[/mathjaxinline] to also be true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_16_04a_2_1">
<fieldset aria-describedby="status_lect_16_04a_2_1">
<div class="field">
<input type="radio" name="input_lect_16_04a_2_1" id="input_lect_16_04a_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_16_04a_2_1-choice_1-label" for="input_lect_16_04a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_16_04a_2_1">
<text> a) [mathjaxinline]f(t)=c(\omega )[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_16_04a_2_1" id="input_lect_16_04a_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_16_04a_2_1-choice_2-label" for="input_lect_16_04a_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_16_04a_2_1">
<text> b) [mathjaxinline]f(t) = \int _{-\infty }^{\infty }d\omega \, c(\omega )e^{-i\omega t}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_16_04a_2_1" id="input_lect_16_04a_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_16_04a_2_1-choice_3-label" for="input_lect_16_04a_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_16_04a_2_1">
<text> c) [mathjaxinline]\int _{-\infty }^{\infty }d\omega \, \left|c(\omega )\right|^{2} = \int _{-\infty }^{\infty }dt\, \left|f(t)\right|^{2}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_16_04a_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_16_04a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_16_04a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Review of derivation" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_16_04a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_16_04a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_04a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_04a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_04a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L26v5" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v5: Uncertainty Principle</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v5">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L26v5" data-block-type="video" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v5: Uncertainty Principle</h3>
<div
id="video_L26v5"
class="video closed"
data-metadata='{"prioritizeHls": false, "sources": [], "autoplay": false, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v5/handler/transcript/translation/__lang__", "speed": null, "generalSpeed": 1.0, "streams": "1.00:iHZTTBklevo", "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "completionPercentage": 0.95, "start": 0.0, "savedVideoPosition": 0.0, "autohideHtml5": false, "saveStateEnabled": false, "recordedYoutubeIsAvailable": true, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v5/handler/publish_completion", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v5/handler/xmodule_handler/save_user_state", "poster": null, "duration": 0.0, "captionDataDir": null, "showCaptions": "true", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L26v5/handler/transcript/available_translations", "ytTestTimeout": 1500, "transcriptLanguages": {"en": "English"}, "completionEnabled": false, "ytMetadataEndpoint": "", "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L26v5"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L26v5">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v5">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/3edce5e609dc96ca825c65f2d1471d79/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L26v5.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_16_ex04b" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26Q6: Applicability of the Uncertainty Principle</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04b" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_16_04b" class="problems-wrapper" role="group"
aria-labelledby="lect_16_04b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_16_04b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04b-problem-progress" tabindex="-1">
Applicability of the Uncertainty Principle
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_04b-problem-progress"></div>
<div class="problem">
<div>
<p>
Choose ALL of the following statements about the uncertainty principle that are TRUE. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_16_04b_2_1">
<fieldset aria-describedby="status_lect_16_04b_2_1">
<div class="field">
<input type="checkbox" name="input_lect_16_04b_2_1[]" id="input_lect_16_04b_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_16_04b_2_1-choice_0-label" for="input_lect_16_04b_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_16_04b_2_1">
<text>a) There exists an uncertainty relation between ANY two variables.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_16_04b_2_1[]" id="input_lect_16_04b_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_16_04b_2_1-choice_1-label" for="input_lect_16_04b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_16_04b_2_1">
<text>b) There exists an uncertainty relation between variables that are Fourier conjugates.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_16_04b_2_1[]" id="input_lect_16_04b_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_16_04b_2_1-choice_2-label" for="input_lect_16_04b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_16_04b_2_1">
<text>c) The uncertainty principle is a general result, independent of quantum mechanics.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_16_04b_2_1[]" id="input_lect_16_04b_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_16_04b_2_1-choice_3-label" for="input_lect_16_04b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_16_04b_2_1">
<text>d) The uncertainty principle only applies to quantum wave functions.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_16_04b_2_1[]" id="input_lect_16_04b_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="lect_16_04b_2_1-choice_4-label" for="input_lect_16_04b_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_16_04b_2_1">
<text>e) The uncertainty principle is just math, but has no physical consequences.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_16_04b_2_1[]" id="input_lect_16_04b_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="lect_16_04b_2_1-choice_5-label" for="input_lect_16_04b_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_16_04b_2_1">
<text>f) The uncertainty principle can be circumvented.</text>
</label>
</div>
<span id="answer_lect_16_04b_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_16_04b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_16_04b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Applicability of the Uncertainty Principle" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_16_04b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_16_04b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_04b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_04b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_04b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_16_ex05" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26Q7: Examples using the Uncertainty Principle</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_05">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3cb69e04686b11ef86190214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_05" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_16_05" class="problems-wrapper" role="group"
aria-labelledby="lect_16_05-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_05" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_05/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_16_05-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_05-problem-progress" tabindex="-1">
Examples using the uncertainty principle
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_16_05-problem-progress"></div>
<div class="problem">
<div>
<p>
Common wireless bandwidths, [mathjaxinline]\Delta f[/mathjaxinline], are [mathjaxinline]20\, \mathrm{MHz}[/mathjaxinline], [mathjaxinline]40\, \mathrm{MHz}[/mathjaxinline], [mathjaxinline]80\, \mathrm{MHz}[/mathjaxinline], and [mathjaxinline]160\, \mathrm{MHz}[/mathjaxinline] (note, this is different from the frequency channels that are used for wireless, which are [mathjaxinline]2.4\, \mathrm{GHz}[/mathjaxinline] and [mathjaxinline]5\, \mathrm{GHz}[/mathjaxinline]). </p>
<p>
Let's assume that the type of data being sent is binary. The data rate is the inverse of the arrival time between bits (ones or zeros). Use the uncertainty principle to calculate the maximum data rate for each bandwidth. Express your answer as a number, which will have units of [mathjaxinline]\mathrm{Mbps}[/mathjaxinline] </p>
<p>
<p style="display:inline">Data rate [mathjaxinline][20\, \mathrm{MHz}][/mathjaxinline] = </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_16_05_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_16_05_2_1" id="input_lect_16_05_2_1" aria-describedby="status_lect_16_05_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_16_05_2_1"/>
<span class="status unanswered" id="status_lect_16_05_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_16_05_2_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">[mathjaxinline]\mathrm{Mbps}[/mathjaxinline]</p>
</p>
<p>
<p style="display:inline">Data rate [mathjaxinline][40\, \mathrm{MHz}][/mathjaxinline] = </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_16_05_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_16_05_3_1" id="input_lect_16_05_3_1" aria-describedby="status_lect_16_05_3_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_16_05_3_1"/>
<span class="status unanswered" id="status_lect_16_05_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_16_05_3_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">[mathjaxinline]\mathrm{Mbps}[/mathjaxinline]</p>
</p>
<p>
<p style="display:inline">Data rate [mathjaxinline][80\, \mathrm{MHz}][/mathjaxinline] = </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_lect_16_05_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_16_05_4_1" id="input_lect_16_05_4_1" aria-describedby="status_lect_16_05_4_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_16_05_4_1"/>
<span class="status unanswered" id="status_lect_16_05_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_16_05_4_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">[mathjaxinline]\mathrm{Mbps}[/mathjaxinline]</p>
</p>
<p>
<p style="display:inline">Data rate [mathjaxinline][160\, \mathrm{MHz}][/mathjaxinline] = </p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_lect_16_05_5_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_16_05_5_1" id="input_lect_16_05_5_1" aria-describedby="status_lect_16_05_5_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_16_05_5_1"/>
<span class="status unanswered" id="status_lect_16_05_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_16_05_5_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">[mathjaxinline]\mathrm{Mbps}[/mathjaxinline]</p>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_16_05_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Examples using the uncertainty principle" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_16_05" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_16_05">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_05-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_05-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_16_05-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>