1.5: The Double Slit Experiment
- Page ID
- 49368
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We now have the tools to model the double slit experiment described above. Far from the double slit, the electrons emanating from each slit look like plane waves; see Figure \(\PageIndex{1}\), where \(s\) is the separation between the slits and \(L\) is the distance to the viewing screen.
At the viewing screen we have
\[ \psi(x=L,t) = A\text{ exp}[i(k_{0}r_{1}-\omega_{0}t)]+A\text{ exp}[i(k_{0}r_{2}-\omega_{0}t)] \nonumber \]
The intensity at the screen is
\[ \begin{align*} |\psi|^{2} &= \{ A \text{ exp}[-i\omega_{0}t](\text{exp}[ik_{0}r_{1}]+(\text{exp}[ik_{0}r_{1}])\}\{A \text{ exp}[-i\omega_{0}t](\text{exp}[ik_{0}r_{1}]+(\text{exp}[ik_{0}r_{1}])^{*} \\[4pt] &= |A|^{2} + |A|^{2}\text{ exp}[i(k_{0}r_{2}-k_{0}r_{1})]+|A|^{2}\text{ exp}[i(k_{0}r_{1}-k_{0}r_{2})]+|A|^{2} \\[4pt] &=2|A|^{2} (1+cos(k(r_{2}-r_{1}))) \end{align*} \nonumber \]
where \(A\) is a constant determined by the intensity of the electron wave. Now from Figure \(\PageIndex{1}\):
\[ r^{2}_{1}= L^{2}+(s/2 -y)^{2} \nonumber \]
\( r^{2}_{1}= L^{2}+(s/2 +y)^{2} \)
Now, if \(y \ll s/2\) we can neglect the \(y^{2}\) term:
\[ r^{2}_{1}= L^{2}+(s/2)^{2}-sy \nonumber \]
\( r^{2}_{1}= L^{2}+(s/2)^{2}+sy \)
Then,
\[ r_{1} \approx \sqrt{L^{2}+(s/2)^{2}}(1-\dfrac{1}{2}\dfrac{sy}{L^{2}+(s/2)^{2}}) \nonumber \]
\( r_{1} \approx \sqrt{L^{2}+(s/2)^{2}}(1+\dfrac{1}{2}\dfrac{sy}{L^{2}+(s/2)^{2}}) \)
Next, if \(L \gg s/2\)
\[ r_{1} \approx L-\dfrac{1}{2}\dfrac{sy}{L} \nonumber \]
\( r_{1} \approx L+\dfrac{1}{2}\dfrac{sy}{L} \)
Thus, \(r_{2}-r_{1} = \dfrac{sy}{L}\), and
\[ |\psi|^{2} = 2|A|^{2} (1+cos(ks\dfrac{y}{L})) \nonumber \]
At the screen, constructive interference between the plane waves from each slit yields a regular array of bright lines, corresponding to a high intensity of electrons. In between each pair of bright lines, is a dark band where the plane waves interfere destructively, i.e. the waves are \(\pi\) radians out of phase with one another.
\[ \dfrac{2\pi}{\lambda}s\dfrac{y}{L}=2\pi \nonumber \]
Rearranging,
\[ y=\dfrac{L\lambda}{s} \nonumber \]