Phy5645/Double pinhole experiment: Difference between revisions

Latest revision as of 13:20, 18 January 2014

(a) As stated in the problem, we assume that the denominators are approximately the same between the two waves. This is justified because the corrections are only on the order of $d/L\!$ , and we are interested in the case where $d<<L\!$ . We require that the numerators have the same phase, namely $kr_{+}-kr_{-}=2\pi n\!$ . We expand the LHS with respect to $d\!$ ,

{\begin{aligned}k(r_{+}-r_{-})&\approx k\left\{\left({\sqrt {L^{2}+x^{2}+y^{2}}}+{\frac {yd}{2{\sqrt {L^{2}+x^{2}+y^{2}}}}}\right)\right.\\&-\left.\left({\sqrt {L^{2}+x^{2}+y^{2}}}-{\frac {yd}{2{\sqrt {L^{2}+x^{2}+y^{2}}}}}\right)\right\}\\&=k{\frac {dy}{\sqrt {L^{2}+x^{2}+y^{2}}}}\end{aligned}}

Therefore, $k{\frac {dy}{\sqrt {L^{2}+x^{2}+y^{2}}}}=2\pi n\!$

and hence

{\frac {y}{\sqrt {L^{2}+x^{2}+y^{2}}}}=n{\frac {\lambda }{d}}\!

(b) Let us work in units in which $k=1$ ; that is, we measure all lengths in units of $1/k\!$ . We then set $d=20\!$ and $L=1000\!$ . Here is the interference pattern. We first plot the pattern along the $y\!$ axis $(x=0):\!$

(c) Using the same parameters as before, we now plot the pattern on the $xy\!$ plane:

(d) We use the same parameters as in (b) and (c). First, we plot the pattern along the $y\!$ axis $(x=0):\!$

We now plot the pattern on the plane:

The main difference is the absence of the interference pattern.

Back to Stern-Gerlach Experiment

@@ Line 1: / Line 1: @@
-Submitted by team 1
+(a) As stated in the problem, we assume that the denominators are approximately the same between the two waves. This is justified because the corrections are only on the order of <math>d/L\!</math>, and we are interested in the case where <math>d<<L\!</math>. We require that the numerators have the same phase, namely <math>kr_{+}-kr_{-}=2\pi n\!</math>. We expand the LHS with respect to <math>d\!</math>,
-------
+:<math>
+\begin{align}
+k(r_{+}-r_{-})&\approx k\left\{\left(\sqrt{L^{2}+x^{2}+y^{2}}+\frac{yd}{2\sqrt{L^{2}+x^{2}+y^{2}}}\right) \right. \\
+&-\left.\left(\sqrt{L^{2}+x^{2}+y^{2}}-\frac{yd}{2\sqrt{L^{2}+x^{2}+y^{2}}}\right)\right\}\\
+&=k\frac{dy}{\sqrt{L^{2}+x^{2}+y^{2}}}
+\end{align}
+</math>
-'''Question: Double Pinhole Experiment'''
+Therefore,<math>k\frac{dy}{\sqrt{L^{2}+x^{2}+y^{2}}}=2\pi n\!</math>
-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.
+and hence
+:<math>\frac{y}{\sqrt{L^{2}+x^{2}+y^{2}}}=n\frac{\lambda}{d}\!</math>
-[[Image:Double_pinhole_1.JPG]]
-Suppose you send in an electron along the z axis on a screen at <math>z = 0</math> with two pinholes at <math>x = 0, y = \pm d/2</math>. On a point <math>(x,y)</math> on another screen at <math>z = L>>d, \lambda,</math> the distance from each pinhole is given by <math> r_{\pm}=\sqrt{x^{2}+(y\mp d/2)^{2}+L^{2}}</math>. The spherical wave from each pinhole is added on the screen and hence the wave function is
+(b) Let us work in units in which <math>k = 1</math>; that is, we measure all lengths in units of <math>1/k\!</math>. We then set <math>d = 20\!</math> and <math>L = 1000\!</math>. Here is the interference pattern. We first plot the pattern along the <math>y\!</math> axis <math>(x = 0):\!</math>
-:<math>\psi(x,y)=\frac{e^{ikr_{+}}}{r_{+}}+\frac{e^{ikr_{-}}}{r_{-}}</math>,
-where <math> k = 2\pi /\lambda\!</math>. Answer the following questions.
+[[Image:Double_pinhole_plot_1.JPG]]
-(a) Considering just the exponential factors, show that the constructive interference appears approximately at
-:<math> \frac{y}{r}=n\frac{\lambda}{d}</math>
+(c) Using the same parameters as before, we now plot the pattern on the <math>xy\!</math> plane:
-where <math> r=\sqrt{x^{2}+y^{2}+L^{2}}</math>.
+[[Image:Double_pinhole_plot_2.JPG]]
-(b) Make a plot of the intensity <math>|\psi(0,y)|^{2}\!</math> as a function of <math>y\!</math>, by choosing <math>k=1, d =20\!</math>, and <math> L=1000 \!</math>. The intensity <math>|\psi(0,y)|^{2}\!</math> is interpreted as the probability distribution for the electron to be detected on the screen, after repeating the same experiment many many times.
-(c) Make a contour plot of the intensity <math>|\psi(x,y)|^{2}\!</math> as afunction of x and y, for the same parameters.
+(d) We use the same parameters as in (b) and (c).  First, we plot the pattern along the <math>y\!</math> axis <math>(x = 0):\!</math>
-(d) If you place a counter at both pinholes to see if the electron has passed one of them, all of a sudden the wave function "collapses". If the electron is observed to pass through the pinhole at <math>y=+d/2\!</math>, the wave function becomes
+[[Image:Double_pinhole_plot_3.JPG]]
-:<math>\psi_{+}(x,y)=\frac{e^{ikr_{+}}}{r_{+}}</math>.
+We now plot the pattern on the plane:
-If it is observed to pass through that at <math>y=-d/2\!</math>, the wave function becomes
+[[Image:Double_pinhole_plot_4.JPG]]
-:<math>\psi_{-}(x,y)=\frac{e^{ikr_{-}}}{r_{-}}</math>.
+The main difference is the absence of the interference pattern.
-After repeating this experiment many times with 50:50 probability for each the pinholes, the probability on the screen will be given by
+Back to [[Stern-Gerlach Experiment#Problem|Stern-Gerlach Experiment]]
-:<math>|\psi_{+}(x,y)|^{2}+|\psi_{-}(x,y)|^{2}</math>

Phy5645/Double pinhole experiment: Difference between revisions

Latest revision as of 13:20, 18 January 2014

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