# Basic QM question

## Homework Statement

Suppose that we have a source of particles (e.g. photons) S, then three slits labelled 1,2 and 3, followed by a screen. For a particle that has passed through slit i, where i=1,2,3, let ψi(x) be the amplitude for the particle arriving at a position x units along the screen.

(a) Write down the probability density function for detecting a particle at a position x on the screen when:

1. all three slits are open,
2. slits 1 and 3 are open,

[Note: you don't need to determine explicit expressions for the amplitudes ψi(x)]

(b) If we applied purely classical physics, how would your answers to the above differ?

(c) Suppose we know the probability density functions for detecting the particle at a position x when only one particular slit is open. That is, we know P1(x), P2(x) and P3(x). Are we able to express the probability density function for the case of all three slits open in terms of P1(x), P2(x) and P3(x)? Can we do this if we apply purely classical physics?

## The Attempt at a Solution

(a) |ψslits 1,2,3|2= (ψ*slit 1 + ψ*slit 2 + ψ*slit 3)(ψslit 1 + ψslit 2 + ψslit 3)

= |ψslit 1|2 + |ψslit 2|2 + |ψslit 3|2 + ψ*slit 1ψslit 2 + ψ*slit 1ψslit 3 + ψ*slit 2ψslit 1 + ψ*slit 2ψslit 3 + ψ*slit 3ψslit 1 + ψ*slit 3ψslit 2

(b) The probability density functions would add linearly in classical physics, giving:

slits 1,2,3|2= |ψslits 1|2 + |ψslits 2|2 + |ψslits 3|2

(c) Well I would just plug the values for P(x) into the above equations for QM and classical physics, respectively, right?

TSny
Homework Helper
Gold Member
For part (c): Is it possible to express P(x) in terms of just P1(x), P2(x), and P3(x)?

Maybe not, because those probabilities don't take into account that the electron could interfere with itself. But it is possible in classical physics because the electron is just a particle.

TSny
Homework Helper
Gold Member
Maybe not, ...

You should be able to give a definite answer by looking at your mathematical expression for the answer to (a). How would you write P1(x) in terms of ψslit 1, etc.?

P1(x) could be written as ψ*slit 1ψslit 1, but my expression of the probability in a) includes terms like ψ*slit 1ψslit 3 which cannot be expressed in terms of P1(x), P2(x) or P3(x). Therefore I cannot express the probability in terms of just P1(x), P2(x) and P3(x). But in classical physics, the probability density functions add linearly, so I could express the probability in terms of P1(x), P2(x) and P3(x).

TSny
Homework Helper
Gold Member
Yes, I think that's what the question wanted.

Great, thanks! :)