Basic QM: Probability Density w/ 3 Slits Open

[phosgene]

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 ψ_i(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 ψ_i(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?

Homework Equations

The Attempt at a Solution

(a) |ψ_{slits 1,2,3}|²= (ψ*_{slit 1} + ψ*_{slit 2} + ψ*_{slit 3})(ψ_{slit 1} + ψ_{slit 2} + ψ_{slit 3})

= |ψ_{slit 1}|² + |ψ_{slit 2}|² + |ψ_{slit 3}|² + ψ*_{slit 1}ψ_{slit 2} + ψ*_{slit 1}ψ_{slit 3} + ψ*_{slit 2}ψ_{slit 1} + ψ*_{slit 2}ψ_{slit 3} + ψ*_{slit 3}ψ_{slit 1} + ψ*_{slit 3}ψ_{slit 2}

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

|ψ_{slits 1,2,3}|²= |ψ_{slits 1}|² + |ψ_{slits 2}|² + |ψ_{slits 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]

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

 

[phosgene]

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]

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 ψ_{slit 1}, etc.?

 

[phosgene]

P₁(x) could be written as ψ*_{slit 1}ψ_{slit 1}, but my expression of the probability in a) includes terms like ψ*_{slit 1}ψ_{slit 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]

Yes, I think that's what the question wanted.

 

[phosgene]

Great, thanks! :)