Something that's bothered me for a while

[Manchot]

In simple systems (e.g., a periodic potential with non-interacting electrons), we speak of the energy eigenstates of the system. If we include secondary effects like spin-spin coupling, we'll often see a splitting of the energy levels into multiple states. Assuming that the amount of splitting is much less than other energy scales in the system, the overall effect would be to increase the degeneracy of each state. Why, then, can we not "fit" more fermions into each state than what the simple theory predicts? For example, if you measure the absorption spectrum of bulk gallium arsenide, you'll find the density of states is approximately what the simple theory predicts, multiplied by 2 for spin. If I didn't know any better, I'd think that all of the second-order effects would increase the degeneracy by several fold.

 

[marcusl]

In simple systems (e.g., a periodic potential with non-interacting electrons), we speak of the energy eigenstates of the system. If we include secondary effects like spin-spin coupling, we'll often see a splitting of the energy levels into multiple states. Assuming that the amount of splitting is much less than other energy scales in the system, the overall effect would be to increase the degeneracy of each state.

No, the splitting breaks the degeneracy of the existing state. The Zeeman and Stark splittings are perfect examples.

Why, then, can we not "fit" more fermions into each state than what the simple theory predicts? For example, if you measure the absorption spectrum of bulk gallium arsenide, you'll find the density of states is approximately what the simple theory predicts, multiplied by 2 for spin. If I didn't know any better, I'd think that all of the second-order effects would increase the degeneracy by several fold.

The splitting doesn't increase the number of states, it just creates new states that are linear combinations of the old ones. Where the original states were degenerate (same energy), the new states are spread apart in energy, but the number of states is unchanged. That's why the occupancy numbers don't change.

 

[Manchot]

Duh, I completely forgot about that. That makes more sense.