Assignment 1

Question 1

1.

• As we know when the photon goes through a beam splitter and is reflected off the silver side it gains a phase shift and not when going through it, so we know the top path gains a phase shift, so we can add another beam splitter that allows the top path to go through and the bottom one goes through the silvered side making it gain a phase shift, destructively interfering with the bottom path, and constructively interfering with the top path, making it go only to detector 2, we can add the phase shifter before to the system to give the same net phase shift.
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2.

• Nearly the same answer as 1, instead there are 2 more phase shifters, but since we have 2 we can put them in parallel or we can put 1 in each to make the net phase shift back to normal and it would always hit detector 2
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3.

• Instead of making it hit only detector 2 we must make it hit only detector 1, we know that the beam splitter adds a phase shift to the top path, using that we must make the bottom path also have an initial phase shift before it goes through the beam splitter, so we can just add a phase shifter to the bottom path, and have the net phase shift for the top path to be none, so we add the last 2 to the top, we also can do the inverse to it and still get the same answer because the destructive interference
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4.

• in order to make it only go to detector 2 we must make the bottom path have destructive interfearnce so to do that we can add a net shift of 0.75pi and then to the dottom a shift of 0.25pi making it destructivly intefere with reflected path
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5.

• in order to make it 50/50 we can just add a 0.25pi phase shift so that way we can add a 0.50pi phase shift to the top path because when the bottom path gets reflected the net phase diffrence is the same as they are both pi/2 so it would be 50/50
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Question 2

• to fined that vector i combined the vector of psi, and then took the dot product of a vector $\vec{v}$ with components $\left[\begin{array}{c}a\\ b\end{array}\right]$ , then using the coefficients i used the normalization formula to find a and b’s specific values
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Question 3

• by just playing around with the simulation i got an experimental value of $\theta$, and going through each matrix representation i used $\theta$ to derive the relation ship of the angle and the probability, using that i constructed the final probability expression and computed the probability
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