Question

Firstly I am sorry for any awkward English expressions.

Recently I'm reading "Feynman Lectures on Physics - Quantum Mechanics" and come to have a single question.

In the book Feynman explains, "You must never add amplitudes for different and distinct final states".

For example) in double slit experiment with two holes (hole 1, hole 2), if we call the probability amplitude of an electron through the hole 1 to an electron detector as Y1 and hole 2 as Y2,

without any disturbing measurments such as a light, we can say, "the probability of electron to reach the detector is [Y1 + Y2]^2 ." and......

however with some light emitting device to distinguish which hole is taken by the electrons - in this case we can know the choice of electrons by photons - the probability of electron to reach the detector is not just [Y1 + Y2]^2 but [Y1]^2 + [Y2]^2.

I do know that the difference between the results depends on the distinguishability of alternatives, but the problem is why distinguishability makes the results different. Why can't the probability amplitudes interfere if we can know which alternative is chosen?

(Additionally, Feynman says, the probabilities of passing hole 1 and hole 2 are independent each other, but I cannot find any relationship between this mention and my problem.)

Answer 1

Why can't the probability amplitudes interfere if we can know which alternative is chosen?

The knowledge of the alternative "chosen" comes with a price; the price of interaction with the electron.

Regardless of the state of the electron before the interaction, the state after the interaction, the interaction with the detection apparatus at either slit, localizes the electron; the wave function "collapses" to a wave packet localized at one slit and unitarily evolving thereafter (until interacting with the detector). Thus there are two distinguishable states and they don't interfere because, due to the localized nature of the wave packet, there is vanishing amplitude at the other slit and so, the probabilities add.

However, if there is no interaction with the electron before it is detected, the state unitarily evolves until detection. The wave function propagates through both slits and so the amplitude is significant at both slits; the electron "takes both paths". The wave function between the slits and detector is essentially proportional to the sum of the two distinguishable states above. When we square this sum of amplitudes to find the probability, there's an interference term.

To summarize, by detecting which path, you "throw away" the other possibility and with it, the interference.