In 1D quantum mechanics problems, the energy spectrum is often determined by the limits of the potential at $\pm$ infinity. Generally, the spectrum is continuous non-degenerate when energy is above both limits, continuous and degenerate when in between, and discrete and degenerate when below both. Often we can even see the transition between the discrete and continuous regimes when we look at, for example, the hydrogen atom. Below $0$, the spectrum is discrete but the levels get infinitely close as you approach positive values and then continuous for the positive values. This is because the potential energy limit on both sides is $0$.

Continuing this investigation, I wanted to know what would happen in potentials without a well defined limit like for example a sinusoidal potential? It seems like a fairly difficult question because there is a distinct qualitative difference between taking the top of the sine as your limit and the bottom so a process of finding the limit of the energy spectrum as you extend the sine wave to infinity seems like it would surely fail. Please let me know.


The canonical name for this problem seems to be 'quantum pendulum'. This is on the edge of solvability, and whether the eigenfunctions are "known", "exact" or "analytical" will depend on your tolerance to highly technical special-functions material. This is because the stationary Schrödinger equation for this system, $$ -{\frac {\hbar ^{2}}{2ml^{2}}}{\frac {\mathrm {d} ^{2}\psi }{\mathrm {d} \phi ^{2}}}+mgl(1-\cos \phi )\psi = E \psi $$ can be tightly mapped to the Mathieu equation, which means that all of the eigenfunctions of the problem are some sort of Mathieu function or other.

On the other hand, it seems that

  1. you're only after qualitative considerations, which makes things easier, and
  2. in contrast to the proper quantum-pendulum problem, you do not want to enforce periodicity in the wavefunctions.

Either way, though, you still have a periodic potential, and this imposes some pretty severe consequences on the eigenstates, by way of the Bloch theorem. Basically, you will have two types of eigenstates:

  • For energies above the peaks of the potential, you have a full continuum of Bloch waves.
  • For energies below the peaks of the potential, you will have states which localize to the classically-allowed regions, but which tunnel through to the next site $-$ coming out with a phase which is governed by the Bloch quasimomentum. Generically, each discrete eigenstate of the periodic quantum pendulum will produce a full band of eigenstates, and as you approach the limit you're quite likely to get a countably infinite set of bands as you approach the peak potential from below.

All in all, though, not a particularly easy problem - and this is for a potential with a huge amount of structure! If you want to look more widely, though, at the class you said you're interested in,

what would happen in potentials without a well defined limit

and you want to explicitly look at non-periodic potentials, then it becomes a much more challenging problem. For that type of potential some results are known (say, along the direction of Anderson localization) but a rigorously-known full spectrum of such a potential is probably too ambitious of a goal.

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  • $\begingroup$ Interesting, so you think it would produce countable infinite continuous bands below the peak? Strange, I assume they are non degenerate though because of the translational symmetry? $\endgroup$ – Aakash Lakshmanan Apr 14 '19 at 22:49

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