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Density of states with respect to energy is often defined as the number of states at a particular energy level. But I find this definition ambiguous. Because according to Pauli exclusion principle no two electrons can have the same state, so at most only 2 electrons can have the same energy. But in semiconductors the density of states for the conduction and valence bands are given by the following equations:

density of states in a semiconductor

which definitely seems to give a finite real number. This is particularly confusing when considered along with the concept of degeneracy. Some sources such as this and this, state that both degeneracy and density of states are all but the same quantities.

If the density of states function does not give the no of states at a particular energy level rather between an interval then what does the fermi function or the maxwell distribution which represent probabilities mean in this context ? It is easier to think about them as the probability that a single state with a particular energy gets filled. But then again according to the Pauli exclusion principle only 2 such states exists.

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Density of states with respect to energy is often defined as the number of states at a particular energy level

This is not the right definition. The density of states $g$ is the function such that the number of states per unit volume with energy in the infinitesimal interval $[E,E+\mathrm dE]$ is given by $g(E) \mathrm dE$. Put differently, the number of states per unit volume with energy between $E_1$ and $E_2$ is $$\frac{N}{V} = \int_{E_1}^{E_2} g(E) \mathrm dE$$

Because according to Pauli exclusion principle no two electrons can have the same state, so at most only 2 electrons can have the same energy.

The last part of that statement is wrong, because you could have a very large number of states which all have exactly the same energy. The Pauli exclusion principle says that each state can be occupied by at most one electron, but if there are a billion states with the same energy then you could have a billion electrons with the that energy.

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  • $\begingroup$ But when two atoms with same energy levels come closer the level split into two, and only 2 opposite spin electons can occuy either of that definite energy levels. The wave function of the electron with that particular energy is delocalised throughout the entire solid. So are you saying that there exists billions of other delocalized wavefunctions with the same energy ? $\endgroup$ Commented Sep 25, 2022 at 16:07
  • $\begingroup$ @Examination12345 No, I’m simply explaining that there may be more than one state with a given energy, so the Pauli exclusion principle does not imply that there are only one or two electrons per energy level. The DOS quantifies this by describing the number of states (per unit volume) in a given energy interval. $\endgroup$
    – J. Murray
    Commented Sep 25, 2022 at 16:33
  • $\begingroup$ isnt all the possible states of a semiconductor represented in the band diagram ? An electron with a particular energy has a particular wavevector k and since the wave function is delocalized no two electrons can have the same wave vector and hence energy. I do not understand where this additional degeneracy is coming from ? $\endgroup$ Commented Sep 25, 2022 at 17:23
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    $\begingroup$ @Examination12345 In the 3D free electron model, the dispersion relation is $E(\mathbf k)=\hbar^2 k^2/2m$. If two wavevectors have the same magnitude, then the corresponding states have the same energy. The set of wavevectors which all have a given nonzero magnitude make up a spherical shell (an infinite number of points), so the degeneracy of every nonzero energy level is infinite. As a result, it isn’t particularly useful to talk about the degeneracy of a particular energy level. $\endgroup$
    – J. Murray
    Commented Sep 25, 2022 at 20:29
  • $\begingroup$ Instead, we talk about the DOS because (i) it is the number of energy levels per unit volume, and is therefore finite, and (ii) because we are basically never concerned with the number of states per unit volume at some specific value of $E$, but rather in an interval $[E,E+dE]$. $\endgroup$
    – J. Murray
    Commented Sep 25, 2022 at 20:31

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