How Many Bound States Exist in a Finite Square Well?

[joker314]

Homework Statement

Based on the finite potential well defined by the following equations, how many bound states are there, which of these states are even and which are odd, and what are their energies?


V(x)= 0 for x$$\leq$$-l/2 and x $$\geq$$ +l/2
V(x)=-$$\hbar$$$$^{2}$$/ma$$^{2}$$

Homework Equations

E=n$$^{2}$$$$\hbar$$$$^{2}$$/2mL$$^{2}$$

The Attempt at a Solution

To find the number of states I set $$\hbar$$$$^{2}$$/ma$$^{2}$$ equal to E=n$$^{2}$$$$\hbar$$$$^{2}$$/2mL$$^{2}$$, substituting the value l in for L to get n$$\leq$$8$$^{1/2}$$

So this tells me that there are 2 bound states. N = 1 is the odd function, and N=2 is the even function. I do not however know how to get their energies, nor do I know if this is the correct way to solve the problem. Do I need to define the hamiltonian in the problem? If so, how do I go about doing that?

 

[nasu]

I am afraid your formula for energy is valid only for an infinite well.
Here you have a finite well (with "depth" = h^2/ma^2).
You have to solve Schrödinger equation for the three zones (x<-1/2, x between -1/ and 1/2 and x>1/2) and then impose boundary conditions. The wave functions from the neighboring regions must match at the boundary. These conditions will provide the allowed energies.
You may need to solve the "matching" equation numerically.
Good luck!