# Why are 2N electrons allowed in an energy band?

In Solid State Physics, there are supposedly 2N electrons allowed in any energy band, with N being the number of atoms in the solid.

Firstly, I understand why there should be N allowed states if there are N atoms present, due to the Pauli exclusion principle.

But why the factor of two? I understand it has something to do with the spin but fail to see how. If I take two seperate Hydrogen atoms with initial states as:

$$\text{H}_1, \text{H}_2 = 1s =\; \upharpoonright \upharpoonright$$

Then, due to the Pauli exclusion principle (No electron may be in an identical state), these can be coupled as so:

$$\text{H}_1 = 1s : \; \upharpoonright (n=1)$$

$$\text{H}_2 = 1s : \; \downharpoonright (n=1)$$

Result:

$$\text{H}_1\text{H}_2 = 1s^2 : \; \downharpoonright \upharpoonright$$

Then this only requires N = 2 seperate electron states, I'm confused to what I'm missing.

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• An electron's state includes its spin. – probably_someone Dec 6 at 18:48
• I'm confused what you mean by this, maybe it was a bad example using Helium as now there are 4 total electron states. But using Hydrogen, which is a metal at low temperature, I have 1 hydrogen atom with spin $\downharpoonright$ and another with $\upharpoonright$ both will be in the $1s$ state, still making N=2 states, not 2N – mojo-3 Dec 6 at 18:51
• Each hydrogen atom has 2 possible states for the electron, one with spin up and another with spin down. Therefore, there are 4 possible states. The $2N$ figure does not refer to the number of states that are occupied, but rather the number of states that can be occupied. – probably_someone Dec 6 at 19:01
• @probably_someone That does make sense, but however due to the Pauli exclusion principle, I believe that the other two states cannot be filled at the same time as the material is now in a band structure, any new electron joining the state would have the same spin as an already present electron? – mojo-3 Dec 6 at 19:18
• Each atom can have many states 1s, 2s, 2p, 3s, 3p, 3d etc etc each with 2 electrons. – PhysicsDave Dec 6 at 19:20