# Adiabatic expansion of infinite square well

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1. May 12, 2016

### Cracker Jack

1. The problem statement, all variables and given/known data

Suppose that an infinite square well has width L , 0<x<L. Nowthe right wall expands slowly to 2L. Calculate the geometric phase and the dynamic phase for the wave function at the end of this adiabatic expansion of the well. Note: the expansion of the well does not occur at a constant rate.

2. Relevant equations

Dynamic phase= -1/ħ * ∫ En(t') dt' from 0 to t
Geometric phase= i * ∫ <ψn,∂t'ψn>dt' from 0 to t

3. The attempt at a solution
First, I calculated the geometric phase by getting the wave function in terms of L and integrating the bra and ket with dL instead of dt. ∂ψ/∂t = ∂ψ/∂L * dL/dt then plugging that in for the geometric phase.

After using the equation for the wave function of the well and some fun integrations the answer for the geometric phase comes out to zero.

However, I am stuck on the dynamic phase.

The energy for a given n is En=n222/(2m*L2)

However, if the expansion doesn't occur at a constant rate the dynamic phase is:

constants*∫ 1/(L+v(t)*t)2 where v(t) is the velocity of the expansion. I'm just not sure how to integrate that to get the dynamic phase.

2. May 15, 2016

### DrDu

This problem occurs every now and then. Nevertheless I find it quite dubious. The change of the boundary will mix in states of infinite energy, so you can't do it so slowly that this effect becomes negligible. There is no proper adiabatic limit.