Suppose I set up the double-slit experiment using photons as my particle. Behind the left slit I place a beam splitter that points some of the light off in the direction of a camera (represented as C in the diagram below). We'll say that the beam splitter splits a light beam passing through it into two equal beams. The light that passes through the beam splitter unaffected and the light that passes through the right slit continue on to another camera (represented as CCC in the diagram below). We'll say that camera C is closer to the beam splitter than camera CCC is.
CCC
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Laser
If I fire a single photon through this apparatus, there's about a $1/4$ chance that it will be detected by camera C and about a $3/4$ chance that it will be detected by camera CCC. When the wavefunction reaches camera C, the photon is forced to either be detected by camera C or have the part of the wavefunction leading to camera C disappear.
Question: In the case that the photon is not detected by camera C, how is the $1/4$ probability from that branch of the wavefunction redistributed to the remaining two branches of the wavefunction, the branch went through the left slit and straight through the beam splitter and the branch that went through the right slit? The first starts with about $1/4$ probability and the second starts with about $1/2$ probability. Does the first get a new probability of $1/3$ and the second a new probability of $2/3$? Does the first get a new probability of $1/2$ while the second keeps its old probability of $1/2$?
Answer
Starting without the beam splitter, just to establish notation, label the slits 1 and 2, the laser L and a point on the screen x. Then $\langle 1| L \rangle$ is the (complex) amplitude to go from the laser to slit 1. $\langle x| 1 \rangle$ is the amplitude to go from slit 1 to a point on the x on the screen. The amplitude to do that path combination is $\langle x| 1 \rangle\langle 1| L \rangle$. Given that it must land somewhere on the screen we know $$ \sum_x(|\langle x| 1 \rangle\langle 1| L \rangle + \langle x| 2 \rangle\langle 2| L \rangle |^2)=1$$ (or integral...)
If we now place a beamsplitter B right after slit 2, then instead we have$$|\langle c| B \rangle \langle B| 2 \rangle \langle 2| L \rangle|^2+\sum_x(|\langle x| 1 \rangle\langle 1| L \rangle + \langle x| B \rangle \langle B| 2 \rangle\langle 2| L \rangle |^2) =1$$ Now we know a few things about the constituents: $$|\langle 1|L \rangle |^2=0.5 $$ $$|\langle 2|L \rangle |^2=0.5 $$ $$|\langle B|2 \rangle |^2=1 $$ $$|\langle c|B \rangle |^2=0.5 $$
So the first term is $\frac{1}{4}$ (propagation into the camera) and the sum of the other terms must be $\frac{3}{4}$ (propagation to somewhere on the screen).
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