# B Energy bands

1. Oct 7, 2016

### bluecap

in energy bands.. is the electron in superposition of energy states (pure states) or is it entangled in some way?

2. Oct 7, 2016

### ZapperZ

Staff Emeritus
Start with the simplest case of tight-binding band structure.

http://lampx.tugraz.at/~hadley/ss1/bands/tightbinding/tightbinding.php

Check for yourself. Look at the wavefunction being constructed in terms of the basis function.

Zz.

3. Oct 7, 2016

### bluecap

I don't understand any of the equation. So is it pure state or in some kind of entangled form?

4. Oct 7, 2016

### ZapperZ

Staff Emeritus
Wait, let's step back a bit. Do you know what an equation of "pure state" or "entangled state" looks like?

Zz.

5. Oct 7, 2016

### bluecap

I only used visualization. Pure state is superposition. Entangled state is extension of superposition or when it can be decomposed to 2 systems. I learnt all this from Bill Hobba.

But in energy bands in solids.. it is very large extend than simple electron in an atom.. so can we still say the energy bands are in pure state or no longer pure?

6. Oct 7, 2016

### ZapperZ

Staff Emeritus
What does this mean? That you only can see drawings of such things?

Wait, what? And you learned this purely (pun intended) from reading PF posts by someone and from nothing else? In other words, if a mathematical form of a superposition of pure states comes along and bite you on the rear end, you will not have been able to identify what it is?

This is puzzling. Being in a "pure" or "mixed" state has nothing to with it not being described by an atom. (scratching head)

Zz.

7. Oct 7, 2016

### bluecap

I read it from Sabine introduction too shared by StevenZ: http://backreaction.blogspot.com/2016/03/dear-dr-b-what-is-difference-between.html

"A “superposition” is simply a sum of two solutions, possibly with constant factors in front of the terms. Now, some equations, like those of quantum mechanics, have the nice property that the sum of two solutions is also a solution, where each solution corresponds to a different setup of your experiment."
"Entanglement is a correlation between different parts of a system. The simplest case is a correlation between particles, but really you can entangle all kinds of things and properties of things. You find out whether a system has entanglement by dividing it up into two subsystems. Then you consider both systems separately. If the two subsystems were entangled, then looking at them separately will inevitably reduce the information. In physics speak, you “trace out” one subsystem and are left with a mixed state for the other subsystem."

An electron in an atom is in superposition of position eigenstates. So I was just asking in solids.. since there are many atoms and molecules.. does the electron in the solid energy bands is also in superposition of position eigenstates? This is all I'm asking.

8. Oct 7, 2016

### ZapperZ

Staff Emeritus
A fundamental issue here is that you somehow are content to learning things from "blogs" and "forums", instead of looking for legitimate sources. The other problem here is that you have this aversion to learning the mathematics.

Aren't you the least bit curious to want to know the mathematical form of these things, so that next time, you can identify the same thing when it occurs once again? In other words, wouldn't you rather that someone teach you how to fish, instead of keep asking for fish each time you need something to eat?

When you posed the question, I had explicitly assumed that you already know what "pure states", "mixed states", and "entangled states" would look like mathematically (and you should be aware that the way you pose your question gave NO indication that you didn't know these things already). It is why I gave that website because it is clear what a superposition of states (pure or not) looks like mathematically.

And no, a superposition need NOT be just from a sum of only two states. This is false. In fact, in many cases, it is an infinite sum of states!

I think you need to go back to the basics first and learn the actual physics (rather than just via "visualization", whatever that means), before attempting to apply these things to solid state physics. After all, that is what all of us in this field had to do.

Zz.

9. Oct 7, 2016

### bluecap

I didn't say a superposition need not be just from a sum of only two states. Didn't I just say the almost infinite position eigenstates of the electron in the atom is in superposition. I read many old archives of Hobba messages so I'm very familiar with their verbal summaries. I'm a layman with just simple understanding of algebra and nothing more (I don't even know calculus). But I know pure state means superposition, Mixed state is when there is a setup that would make it choose of the eigenstates and one outcome. I read Hobba mentioning this a hundred times so I'm familiar with it. Entangement is when two systems interact like system and environmental decoherence. By visualization I mean I use visualization to imagine the molecular bond and quantum orbital shapes and even quantum fields operators. And I know that we must not imagine classical stuff but quantum and relativistic. So I take this into account when visualizing. Now can someone just answer my simple question of whether we can consider energy bands as entanglement of just superposition of the electron in thousand of atoms in the solids? I'm just concerned whether thermal agitations can prevent the solids energy band from pure superposition where the electron energy range were increased due to the very close spacing that form one big wave function or energy bands in the solids.

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