Quantum Mechanincs Infinite well

[benjy1]

Hi so that I can get the help for the specific problem I am working on I will set the question up and include all the steps that I can get and work out. The end question will be about quantized energy levels. This is for a maths module.

I am working on infinite wells and particularly on a question that will tend to the energy levels of an infinite well.

THE PROBLEM

v(x) = infinity x<0 and x>2a , V for a<x<2a and 0 for 0<x<a

and for the energy E>V i can work out the two wave functions

Psi=Asin(kx) for 0<x<a where k^2=2mE/h^2 (i don't know how to put h bar) and Psi= Bsin(K(2a-x)) for a<x<2a where K^2=2m(E-V)/h^2

I use the boundary condition at x=a to find the relation between K and k.

Its that the derivative and the wave functions are equal.

I have

1. Asin(ka)=Bsin(Ka)
2.Akcos(ka)=-BKcos(Ka)

So from this I can get

3. Btan(Ka)/K=-Btan(ka)/k

So now is where I am stuck. I need to solve this where V tends to zero, so this means it will be like an infinite well.
So if V tends to 0 then K tends to k.
so how do I solve 3? I can't cancel the B because B=0 might be a solution.

So what I am thinking is that it is either B=0 or tan(ka)=0

so for tan(ka)=0 we have sin(ka)=0 so k=npi/a

For B=0 then from 1 and 2 we get different solutions. I don't want A=0 so that is ruled out.

I need to find the quantized energy levels. These must be the same as for the infinite well case i.e E=(h*pi*n)^2/8*m*a^2.

To do this k has to be equal to n*pi/2a.

So any advice on what to do further please? Thanks

 

[benjy1]

If any of the people that looked at the problem need any information please let me know.

Any feedback is helpful.