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<h2 class="hd hd-2 unit-title">Introduction to the Uncertainty Principle</h2>
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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>
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<h2 class="hd hd-2 unit-title">L26v1: Review of Fourier transform properties</h2>
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<h2 class="hd hd-2 unit-title">L26Q1: Fourier Transform Example</h2>
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Fourier Transform Example
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Calculate the Fourier transform, [mathjaxinline]c(\omega )[/mathjaxinline], of the following function: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]f(t)=\cos (\omega _{0}t)[/mathjax]</td>
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Choose the correct answer choice below. </p>
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<text> a) [mathjaxinline]c(\omega )= \frac{\sin (\omega _{0}t)}{\omega _{0}}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]c(\omega )= \delta (\omega )[/mathjaxinline]</text>
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<text> c) [mathjaxinline]c(\omega )= \frac{1}{2} \left[\delta (\omega +\omega _{0}) - \delta (\omega -\omega _{0})\right][/mathjaxinline]</text>
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<text> d) [mathjaxinline]c(\omega )= \frac{1}{2} \left[\delta (\omega +\omega _{0}) + \delta (\omega -\omega _{0})\right][/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L26Q2: Comparing f(t) and c(w) [SIMULATION]</h2>
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Visualizing f(t) and c(w)
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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>
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Select ALL true statements that apply, in general, to Fourier pairs. </p>
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<text>a) The "spread" of [mathjaxinline]f(t)[/mathjaxinline] and [mathjaxinline]c(\omega )[/mathjaxinline] are always inversely related</text>
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<text>b) The shape of [mathjaxinline]c(\omega )[/mathjaxinline] will always be a Gaussian function</text>
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<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>
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<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/>
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<h2 class="hd hd-2 unit-title">L26Q3: Calculation of Average Values</h2>
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Calculation of Average Values - part a
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Consider the function introduced earlier in this lesson: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<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>
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<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>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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>
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<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>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<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>
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<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]
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<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<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]
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<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>
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<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>
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<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>
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<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>
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<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>
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Calculation of Average Values - part b
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<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>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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>
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<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>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<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>
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<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]
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
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<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]
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<th class="formulainput" scope="row">Basic functions</th>
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<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]
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<h2 class="hd hd-2 unit-title">Derivation Checkpoint I</h2>
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<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>
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<h2 class="hd hd-2 unit-title">L26Q4: Fourier Identity Proof</h2>
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Fourier Identity Proof
</h3>
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<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">
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<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>
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<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>
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<text> c) [mathjaxinline]\frac{1}{2\pi }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dt'\, dt\, f^{*}(t) f(t') \delta (t-t')[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L26v4: Derivation of Uncertainty Principle</h2>
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<h2 class="hd hd-2 unit-title">Derivation Checkpoint II</h2>
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<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>
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<h2 class="hd hd-2 unit-title">L26Q5: Review of Derivation</h2>
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Review of derivation
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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>
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<text> a) [mathjaxinline]f(t)=c(\omega )[/mathjaxinline]</text>
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<text> b) [mathjaxinline]f(t) = \int _{-\infty }^{\infty }d\omega \, c(\omega )e^{-i\omega t}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\int _{-\infty }^{\infty }d\omega \, \left|c(\omega )\right|^{2} = \int _{-\infty }^{\infty }dt\, \left|f(t)\right|^{2}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L26Q6: Applicability of the Uncertainty Principle</h2>
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Applicability of the Uncertainty Principle
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Choose ALL of the following statements about the uncertainty principle that are TRUE. </p>
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<text>a) There exists an uncertainty relation between ANY two variables.</text>
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<text>b) There exists an uncertainty relation between variables that are Fourier conjugates.</text>
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<text>c) The uncertainty principle is a general result, independent of quantum mechanics.</text>
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<text>d) The uncertainty principle only applies to quantum wave functions.</text>
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<text>e) The uncertainty principle is just math, but has no physical consequences.</text>
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<text>f) The uncertainty principle can be circumvented.</text>
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<h2 class="hd hd-2 unit-title">L26Q7: Examples using the Uncertainty Principle</h2>
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Examples using the uncertainty principle
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<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>
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<p style="display:inline">Data rate [mathjaxinline][20\, \mathrm{MHz}][/mathjaxinline] = </p>
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<p style="display:inline">[mathjaxinline]\mathrm{Mbps}[/mathjaxinline]</p>
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<p style="display:inline">Data rate [mathjaxinline][40\, \mathrm{MHz}][/mathjaxinline] = </p>
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<p style="display:inline">[mathjaxinline]\mathrm{Mbps}[/mathjaxinline]</p>
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<p style="display:inline">Data rate [mathjaxinline][80\, \mathrm{MHz}][/mathjaxinline] = </p>
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<p style="display:inline">[mathjaxinline]\mathrm{Mbps}[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\mathrm{Mbps}[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Descriptions</th>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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>
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<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>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<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>
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<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]
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<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<code>abs, ln, sqrt</code>
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<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]
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<code>sin, cos, tan, sec, csc, cot</code>
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<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>
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<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>
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<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>
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<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>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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