Separating the hamiltonian for a superlattice -- is it this easy? I've been banging my head against a wall trying to figure out what I'm sure is a very simple problem. I want to solve the Kronig Penney model for a superlattice, which is just a normal periodic 1D potential, except the potential wells extend into all space in the directions perpendicular to the perodicity, like so:

I know how to solve a normal 1D KP model. In extending it to 3D, the wave function is now $\psi(r,z) = \psi(z) \psi(r)$, where $r = \{x,y\}$. In the x and y directions, the wave functions are just plane waves, so you can write $\psi (r) = A e^{i\vec k \cdot \vec r}+B e^{-i\vec k \cdot \vec r}$.So, the Hamiltonian applied to the wavefunction gives:
$$\frac{-\hbar^2}{2m}[\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial x^2}+\frac{\partial ^2}{\partial z^2}]\psi(z) \psi(r) + V(z)\psi(z) \psi(r) = E\psi(z) \psi(r)$$
and
$$\frac{\hbar^2 k_\parallel^2}{2m}\psi(z) \psi(r) + \psi(r)\frac{-\hbar^2}{2m}\frac{\partial ^2}{\partial z^2}\psi(z) + V(z)\psi(z) \psi(r)= E\psi(z) \psi(r)$$
(where $k_\parallel = \{k_x,x_y\}$.)
But now is it really so simple to just divide the whole thing by $\psi(r)$ and subtract the parallel kinetic energy terms, and define a new $z$ energy?
$$\frac{-\hbar^2}{2m}\frac{\partial ^2}{\partial z^2}\psi(z) + V(z)\psi(z)= (E-\frac{\hbar^2 k_\parallel^2}{2m})\psi(z) = E_z \psi(z)$$
This is just the regular old KP model problem, but with an energy that has two parameters. So for an electron traveling at an angle away from the normal, once you specify that angle, it's just the 1D problem again. Is it this simple or am I missing something?
Thank you!
 A: I'm don't have high enough reputation to comment yet, so here's a half answer - half comment...
It is indeed possible to separate the problem, as you say, into $\psi(x,y,z) = \chi(x,y)\zeta(z)$ and the total energy of the state is then
\begin{equation}
  E = E_z + \frac{\hbar^2 k_{\parallel}^2}{2m^*_{\parallel}}
\end{equation}
Now, read on if you would like to make the problem a little harder to solve... :-p
Since your diagram refers to a semiconductor heterostructure, you need to use the effective mass of the appropriate energy band rather than the free-electron mass.  Also, if you want your model to be accurate, you must account for the large mismatch in effective mass between GaAs and AlAs.  In other words, the effective mass is a function of $z$, and can be denoted $m^*(z)$.  This should be accounted for first when you find $E_z$ and $\psi_z(z)$ using an appropriate solution to the Kronig-Penney model.  See for example, P. Harrison, "Quantum Wells, Wires and Dots" 3rd Ed., Wiley (2009) if your model doesn't account for this effect yet.
Next, you need to find an appropriate in-plane effective mass that accounts for the spatial variation in $m^*(z)$.  A reasonable approach would be to take the expectation value of the effective mass in the growth direction... this, then depends on which $\psi_z(z)$ state your in-plane dispersion is based upon:
\begin{equation}
  \frac{1}{m^*_{\parallel}(E_z)} = \left\langle\psi_z\right|1/m^*(z)\left|\psi_z\right\rangle
\end{equation}
Therefore, the complete solution would be to (i) find the $z$-component (ii) compute the expectation value of the effective mass (iii) use this to compute the $r$-component.
If you want to make things even harder, then note that the energy bands in a semiconductor are not parabolic, and hence the effective mass will become a function of both energy and $z$-position.  This makes the Schroedinger equation a nonlinear eigenvalue problem, which would be even more challenging to solve... I don't know if there is an analytical solution in that case!
