3d motion of electron in 1d quantum well

Question

Quantum wells where the potential varies along only one direction are typically solved assuming that electrons can only move along the direction. What happens when the 'well' is actually an infinite slab and the electron is free to move in any direction? The wavefunction can be broken up as X(x)Y(y)Z(z). If the well potential is defined along x, then I would I be correct to assume the following:

Y and Z are arbitrary free particle wavefunctions.
Since free particle eigenmodes are not normalizable, it is impossible to have a definite energy state for the total wavefunction.
The energy calculated in any experiment will always be greater than the ground state energy contribution from X(x).
If we plot the wavevector $k = sqrt{k_y^2 + k_z^2}$ against $E_{total}$ we would see a typical band structure diagram with well separated modes and a continuum of states in the case of a finite well.

Is this sort of device used at all in physics? Any insight would be appreciated, thanks.

Artem Alexandrov · Accepted Answer

The ansatz for wave-function is
$$psisim e^{ik_{|}x_{|}}f(z),$$
where $k_{|}=(k_x,k_y)$. The energy can be written as
$$E=frac{hbar^2k_{|}^2}{2m}+frac{k_{z,n}^2}{2m}.$$
It seems that the energy is quite definite, but the energy values are not discrete. It is similar to the particle in magnetic field (the problem about Landau levels). If we plot $E=E(k_{|})$ for different valeus of $n$ (assuming $k_{z,n}sim n^2$), we see band structure,

The following dispersion law appears in different systems. Briefly, you can check papers that relate to:

Different effects in heterojunctions, where one of layers has finite thickness
Effects in external magnetic field (Landau levels reproduces mentioned dispersion law)

Heterojunctions are widely using in a lot of devices, so I assume it will be relatively easy to find something.

3d motion of electron in 1d quantum well

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