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<h2 class="hd hd-2 unit-title">Introduction to N-slit Interference and Other Phenomena</h2>
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Last week, we studied the properties of two-slit interference. Motivated by trying to understand how phased-array radar works, we now look at three-slit interference. Ultimately, we extrapolate to N-slit interference. </p><p>
Additionally, we consider the interference behavior of single electrons. Do single electrons behave like particles or waves? </p>
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<h2 class="hd hd-2 unit-title">L38v1: Three Slit Interference</h2>
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<h3 class="hd hd-2">L38v1: Three Slit Interference</h3>
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<h2 class="hd hd-2 unit-title">L38Q1: Adding More Slits</h2>
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Adding More Slits - part a
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Consider a plane wave with wavelength [mathjaxinline]\lambda[/mathjaxinline] that is transmitted through a grating with [mathjaxinline]N[/mathjaxinline] slits, where the separation between neighboring slits is [mathjaxinline]d[/mathjaxinline]. The interference pattern is projected on a screen a distance [mathjaxinline]L[/mathjaxinline] very far away from the grating (i.e., [mathjaxinline]L\gg Nd)[/mathjaxinline]. The angle [mathjaxinline]\theta[/mathjaxinline] characterizes the position of interference features on the distant screen, relative to the line that bisects the middle of the grating. </p>
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<p><b class="bfseries">(Part a)</b> Which of the following conditions enables the small angle approximation to be made? </p>
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<text> c) [mathjaxinline]\dfrac {\lambda }{d} \approx 1[/mathjaxinline]</text>
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Adding More Slits - part b
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<p><b class="bfseries">(Part b)</b> Determine the width of the central maximum, using the small angle approximation, where the variable [mathjaxinline]y[/mathjaxinline] characterizes the position on the observing screen. Hint, this will be the difference between the location of the first minimum on either side of the central maximum. Express your answer using <code>N</code>, <code>L</code>, <code>d</code>, and <code>lambda</code> for [mathjaxinline]\lambda[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\Delta y_{central} =[/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">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>
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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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<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">L38v2: N-slit Interference</h2>
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<h2 class="hd hd-2 unit-title">L38Q2: N-slit Derivation</h2>
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N-slit Interference
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We asserted that the phase difference between two light rays from nearest neighbor slits hitting the same spot on a screen, [mathjaxinline]\delta _{\mathrm{nn}}[/mathjaxinline], was constant. Consequently, the phase difference between two light rays from the [mathjaxinline]N^{\mathrm{th}}[/mathjaxinline] slit and the first slit is: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\delta _{1N}=(N-1)\delta _{\mathrm{nn}}[/mathjax]</td>
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What condition(s) are necessary for this to hold? Select ALL that apply. </p>
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<text>b) The separation between the slits is small compared to the distance to the screen.</text>
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<text>c) The total spatial extent of the grating (all of the [mathjaxinline]N[/mathjaxinline] slits) is small compared to the distance to the screen, such that all paths are approximately parallel.</text>
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<h2 class="hd hd-2 unit-title">L38v3: N-slit Interference Intensity</h2>
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<h2 class="hd hd-2 unit-title">L38Q3: Varying Number of Slits</h2>
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Varying N - part a
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Consider a plane wave with wavelength [mathjaxinline]\lambda[/mathjaxinline] that is transmitted through a grating with [mathjaxinline]N[/mathjaxinline] slits, where the separation between neighboring slits is [mathjaxinline]d[/mathjaxinline]. The interference pattern is projected on a screen a distance [mathjaxinline]L[/mathjaxinline] very far away from the grating (i.e., [mathjaxinline]L\gg Nd)[/mathjaxinline]. An example of an interference pattern is shown below. Assume the small angle approximation so that [mathjaxinline]L\gg y[/mathjaxinline]. </p>
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<p><b class="bfseries">(Part a)</b> The interference pattern exhibits "large" maxima, and "small" maxima. When the number of slits increases, but the slit separation remains the same, what happens to the locations of the "large" maxima. </p>
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<text> a) the locations get farther apart</text>
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<text> c) the locations do not change</text>
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Varying N - part b
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<p><b class="bfseries">(Part b)</b> The intensity as a function of the phase difference between adjacent slits, is: </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]\langle I \rangle = I_{0} \dfrac {\sin ^{2}\left( \frac{N\delta }{2}\right)}{\sin ^{2}\left( \frac{\delta }{2}\right)}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
What is the intensity distribution for two-slit interference, [mathjaxinline]N=2[/mathjaxinline]? </p>
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<text> a) [mathjaxinline]\langle I \rangle = 2 I_{0} \sin ^{2}\left( \frac{\delta }{2} \right)[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\langle I \rangle = 2 I_{0} \sin ^{2}\left(\delta \right)[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\langle I \rangle = 4 I_{0} \sin ^{2}\left( \frac{\delta }{2} \right)[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\langle I \rangle = 4 I_{0} \sin ^{2}\left(\delta \right)[/mathjaxinline]</text>
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<text> e) [mathjaxinline]\langle I \rangle = 2 I_{0} \cos ^{2}\left( \frac{\delta }{2} \right)[/mathjaxinline]</text>
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<text> f) [mathjaxinline]\langle I \rangle = 2 I_{0} \cos ^{2}\left(\delta \right)[/mathjaxinline]</text>
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<text> g) [mathjaxinline]\langle I \rangle = 4 I_{0} \cos ^{2}\left( \frac{\delta }{2} \right)[/mathjaxinline]</text>
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<text> h) [mathjaxinline]\langle I \rangle = 4 I_{0} \cos ^{2}\left(\delta \right)[/mathjaxinline]</text>
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Varying N - part c
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<p><b class="bfseries">(Part c)</b> What phase shift will generate the [mathjaxinline]n^\mathrm {th}[/mathjaxinline] "large" maxima in an interference pattern generated by a grating with [mathjaxinline]N[/mathjaxinline] slits, where [mathjaxinline]n=0[/mathjaxinline] corresponds to the central maximum? Note: It may be useful to consider your answers from part (a) and (b). </p>
<p>
Express your answer in terms of <code>n</code> and/or <code>N</code> as needed (and any relevant numerical constants). </p>
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<p style="display:inline">[mathjaxinline]\delta =[/mathjaxinline] </p>
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<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>
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<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>
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<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>
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<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>
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<h2 class="hd hd-2 unit-title">L38Q4: N slits [WITH SIMULATION]</h2>
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N Slits - part a
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<p><b class="bfseries">(Part a)</b> For each of the following plots, determine the number of slits, [mathjaxinline]N[/mathjaxinline], that generated the interference pattern shown. Note, the intensities are normalized to a max value of 1. </p>
<p><b class="bfseries">(Part i)</b> Figure 1. </p>
<p>
<p style="display:inline">[mathjaxinline]N =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part ii)</b> Figure 2. </p>
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<p style="display:inline">[mathjaxinline]N =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part ii)</b> Figure 3. </p>
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<p style="display:inline">[mathjaxinline]N =[/mathjaxinline]</p>
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N Slits - part b
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<p><b class="bfseries">(Part b)</b> If the separation between adjacent slits is decreased, how will this affect the interference pattern? Select ALL that apply. </p>
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<text>a) The distance between principle maxima will increase</text>
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<text>c) The distance between principle maxima will stay the same</text>
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<text>d) The width of the principle maxima will increase</text>
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<text>e) The width of the principle maxima will decrease</text>
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<text>f) The width of the principle maxima will stay the same</text>
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<text>g) The number of secondary maxima will increase</text>
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<text>h) The number of secondary maxima will decrease</text>
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<text>i) The number of secondary maxima will stay the same</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=Nslit_interference_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.
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<h2 class="hd hd-2 unit-title">L38Q5: Phased Radar</h2>
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Phased Radar
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Again, consider a plane wave with wavelength [mathjaxinline]\lambda[/mathjaxinline] that is transmitted through a grating with [mathjaxinline]N[/mathjaxinline] slits, where the separation between neighboring slits is [mathjaxinline]d[/mathjaxinline]. The interference pattern is observed a distance [mathjaxinline]L[/mathjaxinline] very far away from the grating (i.e., [mathjaxinline]L\gg Nd)[/mathjaxinline]. Assume the small angle approximation so that [mathjaxinline]L\gg y[/mathjaxinline]. </p>
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<p><b class="bfseries">(Part a)</b> What is the condition on the phase shift between adjacent slits [mathjaxinline]\delta[/mathjaxinline] for constructive interference that produces the first principle maximum which is not at [mathjaxinline]y=0[/mathjaxinline]? Enter your answer as a positive number. </p>
<p>
<p style="display:inline">[mathjaxinline]\delta =[/mathjaxinline] </p>
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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>
</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]
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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>
<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>
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<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>
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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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<p><b class="bfseries">(Part b)</b> Now we wish to shift the position of that first principle maximum by a distance [mathjaxinline]\Delta[/mathjaxinline] from its original location at [mathjaxinline]y_{1}[/mathjaxinline] by adding a phase shift [mathjaxinline]\phi[/mathjaxinline] between adjacent slits. One way to add such a phase shift is by having the incoming wave at an angle to the line of slits, as shown below, since in this case the wavefront will reach adjacent slits at different times. However, the details of how the phase shift is generated is not important, all you need to know is that there is a slit-to-slit phase shift of [mathjaxinline]\phi[/mathjaxinline]. </p>
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<p>
Determine the value of [mathjaxinline]|\phi |[/mathjaxinline] that will produce this change in the location. Express your answer using <code>L</code>, <code>d</code>, <code>lambda</code> for [mathjaxinline]\lambda[/mathjaxinline], and <code>Delta</code> for [mathjaxinline]\Delta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\phi | =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Allowable Entries</th>
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</tr>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
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<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
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<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>
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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">L38v5: Single Electron Interference</h2>
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<h2 class="hd hd-2 unit-title">L38Q6: Single Electron Wavefunction</h2>
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Single electron interference
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Let's consider the single-electron interference experiment described previously. A source emits electrons towards an aperture with two slits. Which of the following observations are true? Select all that apply. </p>
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<text>a) When more than one electron is emitted at a time, the interference pattern is destroyed and Gaussian-like distributions will be observed</text>
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<text>b) When one of the slits is covered, the two-slit interference pattern is destroyed and a single Gaussian-like distribution will be observed</text>
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<text>c) When one of the slits is covered, the two-slit interference pattern changes to a single-slit diffraction pattern</text>
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