An atom laser spits out a pulse of atoms that in some ways behaves like light fr

Question

An atom laser spits out a pulse of atoms that in some ways behaves like light from an optical laser, in the sense that it has a well defined wavelength. When running our atom laser in pulsed operation, each atom in the pulse is described by the initial wavefunction

2
?(x,0)=Ae2?2 e , (1)

where k0 and ? are determined by how we pulse our atom laser. Assume that there are no forces acting on our atoms, that is V (x) = 0.

1. (1 mark) Calculate the momentum space wavefunction. What is the expectation value of the momentum? You can calculate it if you want, but it

Explanation / Answer

E=12kx2

KE=12kx2sin2(?ot+?)

PE=12kx2cos2(?ot+?)

Would suggest at

x=0

would have a minimum and where

cos2(?ot+?)=0

would be another minimum.

For a max we have

x=?

with

cos2(?ot+?)=1

.

We are looking for the bounds of the KE. Since 0?sin2??1, the KE is bounded by 0 and E. Now we plug in the E for the ground state of the quantum harmonic oscillator, which is ...?

p=mdxdt

. So I guess I'm not quite sure what you mean. I'm probably just being an idiot here and it should be very simple.

Since KE = p^2/2m, the bounds on KE tell us the bounds on p for the classical oscillator.

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