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I have not found a good book on statistical mechanics that explains the quantity density of states well. The books I have read so far make the continuum limit approximation, which does not make much sense to me still and the calculation for particles in 1,2,3D is carried out. Please refer me to a book that delves a bit deeper into the idea of density of states. To me I cannot get behind the idea of thinking of the number of states as a function of energy is a continuum, the whole point of Quantum mechanics is that energy states are discrete.

realanswers
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It's a matter of calculus, the whole point is that sometimes it is easier to carry out an integral than a sum.

The sum over states might be replaced by a sum over energies bearing in mind the degeneracy of the energy levels. Sums over energy levels can be approximated by integrals under some circumstances.

$\sum_{states}=\sum_E \Omega(E)\approx\int_E dE \ g(E) $

Basically, the only requirement is that the energy levels are "close enough" so they can be approximated by a continuum. For example, this method can be applied in systems in which the spectrum scales with the inverse of some extensive parameter(e.g. $\Delta E \sim 1/V $).

The computation of the grand-canonical partition function for the ideal gas of bosons is particularly illustrative. The approximation for an integral is needed, yet you have to take some care about how to treat the ground state.

Any classical book discusses this matter. This book could be of help:

Baus, M., & Tejero, C. F. (Eds.). (2008). Equilibrium statistical physics: phases of matter and phase transitions. Berlin, Heidelberg: Springer Berlin Heidelberg.

Edit @LucasBaldo points out, that the continuum limit becomes actually become exact in some cases, such as when the system size is infinite or we deal with Bloch states in crystals.

Javi
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    I would also add that energy levels don't need to be discrete in QM, as long as you allow your system to be infinite in size. In condensed matter, for example, the energy levels are usually parametrized by a variable $k$ which is continuous for an infinite, periodic system. – Lucas Baldo May 09 '22 at 06:17
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    Nice comment. I am not familiar with these kind of systems, could you point me out a reference? I would add it as a reference. – Javi May 09 '22 at 10:20
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    The parameter $k$ I was refering to comes from Bloch's Theorem, which allows one to talk about things like band structures. These topics are covered in standard Condensed Matter textbooks such as "Introduction to Solid State Physics" by Kittel, and "Solid State Physics" by Ashcroft and Mermin. – Lucas Baldo May 09 '22 at 15:13