Light as a Probability Wave — Question 6

Question 6

Discrete detection amplitudes require normalization before conditioning

A photon is in three mutually exclusive orthonormal detection modes with amplitudes $A(1,2i,-1)$. Take A real and positive. Detectors distinguish all three modes perfectly.

1. Normalize the state and find the three detection probabilities.

2. Given that mode 2 did not click, find the conditional probability of mode 3.

Question 6 - Solution

Discrete detection amplitudes require normalization before conditioning

(a) Method and calculation

$$1=|A|^2(1+4+1)\quad\Rightarrow\quad\boxed{A=1/\sqrt6}.$$$$\boxed{(P_1,P_2,P_3)=(1/6,2/3,1/6)}.$$

(b) Interpretation and further calculation

$$P(\text{not }2)=P_1+P_3=1/3.$$$$\boxed{P(3\mid\text{not }2)=\frac{P_3}{P_1+P_3}=1/2}.$$

Physical check

The minus sign and the factor i matter if the modes are later mixed coherently; they do not change probabilities of this direct mode-resolving measurement.