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Perturbation of a negative potential time dependent problem

Hello,

I am trying to calculate the probability that the n = 3 state will be excited at t = infinity for the case where b<<< a. and to show that for any value of b only the odd n states will be excited at t = infinity.

My initial conditions and setup are:

V(x,t) = Vinit(x) exp(-lambda times t) where I have a negative potential in the center of an infinite square well for the case where the perturbation fades away exponentially in time.

I set my potential at -V in the center between the -a and a boundaries and the perturbation extends from -b to +b in the center so the pertubation is as follows:

Vinit(x) = 0 for -a < or = x , -b
Vinit(x) = -V for -b, or = x , or = b
Vinit(x) = 0 for = b, x, or = +a

the sq well is like this: | |
| |
|_____ _____|
-a -b | | b a
|___|

And my unperturbed wavefunctions between a and -a are:

square root of 1/a sin(n*pi*x/2*a) for n even, and same only cos for n odd

I hope this gives enough detail for you. I have been working on this one but am having trouble with the t = infinity and b <<< a conditions.

Solution Preview

Okay, I think you have some problems with some of your assumptions.

1) The equation of state for the unperturbed state is not as you claim.

The example you need to work with is a bound state in a square well
here x = -b and x =b as per your diagram and
and V(x) = -Vo -b<x<b
0 for |x|>b

I set my potential at -V in the center between the -a and a boundaries and the perturbation extends from -b to +b in the center so the perturbation is as follows:

Vinit(x) = 0 for -a < or = x , -b
Vinit(x) = -V for -b, or = x , or = b
Vinit(x) = 0 for = b, x, or = +a

Without the part concerning the cutoff at a and -a this is the case of the square-well potential where over time the depth of the potential is reduced forcing the higher energy state particles out of the trap and only retaining the low energy states until finally as t gets large enough all states degenerate to ...

Solution Summary

This solution is provided in 737 words. It discusses how to solve the problem using perturbation theory and Euler's identiy.

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