Phy5645/Double pinhole experiment

Submitted by team 1

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Question: Double Pinhole Experiment

Besides Stern-Gerlach experiment, double slit experiment is another one that demonstrates how different quantum mechanics is from the classic counterpart. To avoid mathematical complications with Bessel function, we discuss two pinholes rather than slits.

Suppose you send in an electron along the z axis on a screen at $z=0$ with two pinholes at $x=0,y=\pm d/2$. On a point $(x,y)$ on another screen at $z=L>>d,\lambda ,$ the distance from each pinhole is given by $r_{\pm }={\sqrt {x^{2}+(y\mp d/2)^{2}+L^{2}}}$. The spherical wave from each pinhole is added on the screen and hence the wavefunction is

$\psi (x,y)={\frac {e^{ikr_{+}}}{r_{+}}}+{\frac {e^{ikr_{-}}}{r_{-}}}$,

where $k=2\pi /\lambda \!$. Answer the following questions.

(a) Considering just the exponential factors, show that the constructive interference appears approximately at

${\frac {y}{r}}=n{\frac {\lambda }{d}}$

where $r={\sqrt {x^{2}+y^{2}+L^{2}}}$.

(b) Make a plot of the intensity $|\psi (0,y)|^{2}\!$ as a function of $y\!$, by choosing $k=1,d=20\!$, and $L=1000\!$