Light as a Probability Wave — Question 1

Question 1

A lossless mixer converts a relative phase into complementary probabilities

A single photon enters a two-path mixer with normalized amplitudes $a=1/\sqrt2$ and $b=e^{i\phi}/\sqrt2$. Output amplitudes are $c_+=(a+b)/\sqrt2$ and $c_-=(a-b)/\sqrt2$. Detectors are ideal.

1. Derive both output probabilities and check normalization.

2. Find the probabilities at $\phi=\pi/3$ and explain what one detection records.

Question 1 - Solution

A lossless mixer converts a relative phase into complementary probabilities

(a) Method and calculation

$$c_\pm=\tfrac12(1\pm e^{i\phi}),\quad P_\pm=|c_\pm|^2.$$$$\boxed{P_\pm=\tfrac12(1\pm\cos\phi)},\quad P_++P_-=1.$$

(b) Interpretation and further calculation

$$\phi=\pi/3:\quad\boxed{P_+=3/4,\qquad P_-=1/4}.$$$$\boxed{(n_+,n_-)\in\{(1,0),(0,1)\}\quad\text{for one photon}}.$$

Physical check

Probability amplitudes combine before taking the absolute square. A fractional probability is an ensemble prediction, not a fraction of a photon deposited at each ideal detector.