What is the 3D wave function and energy of an electron in an extended 1D well?

[sgod88]

1. Quantum well structure can be realized by sandwiching layers of semiconductor and two insulators.
This sandwiching problem was often treated with 1 D infinite well. Suppose now the problem is 3 dimensional well with length L at z direction from 0 to L, at which

V(x,y,z)= 0 when 0<z<L
infinity otherwise
where V(x) and V(y) is 0. We assume x and y be infinitely large.

Wht is the total energy and the wave function of the electron in such well?
2. I have done the separation of varibales in the Schrödinger equation and obtained the three independent wavefunction.
$$-\frac{\hbar^{2}}{2m}\psi_{x_{i}}=E \psi_{x_{i}}$$
But i don't know what is the boundary condition of the x and y.
I only got psi(z) is the psi of the one d wavefunction.
$$\psi_{z}=\sqrt{\frac{2}{L}}sin(\frac{n \pi z}{L})$$
I just cannot get the constant for the wavefunctions for x and y.
I know that
$$\psi(x)=\psi(x+2\pi)$$
but I still cannot get the value of the constant and the energy.

 

[xepma]

Why do you assume periodic boundary conditions?

It is an infinite well. Remember, that means the wavefunction is zero at all boundaries of the well.

 

[sgod88]

But now V(x,y,z)=V(z), where V is not a function of x and y anymore.
With this condition, in x and y, the wavefunction is certainly not zero, it is a free particle.
$$\psi_{x}=e^{i(kx-wt)}$$