Consider an electron in a periodic potential V(x) such that V(x+a) = V(x) for some real number a. The energy eigenstates are obtained from time-independent SE, which in atomic units is(adsbygoogle = window.adsbygoogle || []).push({});

[itex]-\frac{1}{2}\frac{\partial^{2}\psi(x)}{\partial x^{2}}+V(x)\psi(x)=E\psi(x)[/itex]

According to Bloch theorem, the energy eigenstates are of form [itex]\psi(x)=exp(ikx)\phi(x)[/itex]

where [itex]\phi(x)[/itex] has the same periodicity as V(x).

If at least one eigenfunction is known, the potential V(x) can be solved from the SE with an inverse formula:

[itex]V(x)=\frac{\psi''(x)}{2\psi(x)}[/itex]

Here the eigenvalue E has been arbitrarily chosen to be zero (changing its value only corresponds to adding a constant term to V(x). Plugging the expression for Bloch wavefunction in this equation and differentiating, we get

[itex]V(x)=-k^{2}+ik\frac{\phi'(x)}{\phi(x)}+\frac{\phi''(x)}{2\phi(x)}[/itex]

From this equation one can easily see that the only way how a real-valued [itex]\phi(x)[/itex] can correspond to a real-valued potential V(x) is that [itex]\phi(x)[/itex] is the trivial constant function. Therefore, in most Bloch wavefunctions that correspond to a physically possible potential, [itex]\phi(x)[/itex] is a complex-valued function.

Questions: Why is the range of physically possible Bloch wavefunctions so limited? What's the simplest way to express the minimal condition for function [itex]\phi(x)[/itex] that guarantees real-valued V(x) ? Can anyone give even one nontrivial example of a (differentiable) Bloch-type wavefunction that corresponds to a real potential.

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# Possible Bloch wavefunctions

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