When can the time-independent Schrodinger be used?

[sokrates]

Hi,

I am new to this forum. I realized that I was browsing the forums way too much and I said why not register and post some questions that have been lingering in my head:

Here is one:

In a periodic solid, we almost always neglect the time factor of the Schrödinger equation:

$$e^{-i Et/hbar}$$

So I guess the implicit assumption here is that there's no energy exhcange with the environment (no phonons, etc...) so that the total energy of the electron remains the same?
But what about the potential landscape the electron sees? The potential due to lattice atoms is changing periodically.

So the potential energy $$U(x)$$ in the Schrödinger equation is changing.

How can we understand this then?

 

[Ben Niehoff]

If the potential U(x) is a function of x alone (and not time), then Schrödinger's equation separates, and we may use the the time-independent version to analyze the spatial part of the wavefunction.

 

[sokrates]

I guess we should add that "if the electron is moving $$\textbf{coherently}$$ through the lattice..."

But that something we almost always assume, right?

Thank you for the response.