Why should we use quantum mechanics to study solid state physics and not classical mechanics? I have seen many approaches to studying solid state physics and many of them use quantum mechanics, but why do we use it and not classical mechanics?
 A: Quantum mechanics is used to analyze the inner workings not only of solids, but every other phase state, like liquids, gas, plasma and even bose-Einstein condensate. The reason behind is that at the microscopic level (or sometimes at macroscopic as well, as in case of BEC),- classical physics fails to describe behavior of system. For example classical electromagnetism can't explain discrete emission spectrum of elements.
A: Answerers who've said "because that's how you match experimental reality" are right - that is, in the end, the most important test of any physical model.
However, there is also a theoretical answer: for classical mechanics to be a decent approximation, one needs the length scale on which the wavefunction varies to be small compared with the length scale on which the potential varies.  Although the Pauli exclusion principle requires the electrons to stack up (the Fermi liquid) into states with quite small wavelengths, even the shortest-wavelength ones (the Fermi surface) still have a length scale of variation of the wavefunction that's of the same order as, not small compared with, the size of the unit cell of the material.  The potential varies on the length scale of the unit cell (and possibly on some smaller length scales too), so the condition for classical mechanics to be OK is not met.
A: Quantum mechanic was motivated (in part) by solid state physics. It is often said that the ultraviolet catastrophe was the main issue with classical physics. But, arguably, the first issue was that of the heat capacity (and equipartition). The black-body started to be a matter of concern, during 1895 I think. But the heat capacity troubled the physicists since 1885...
In classical mechanics, each quadratic degree of freedom of a system contributes to the energy: $\frac{1}{2}kT$. Since the heat capacity is: $$C_v = \frac{\partial E}{\partial T}$$
it's an easy thing to calculate from the energy. Now, in diatomic gases, we have a lot of degree of freedom which are quadratic. The translation energy ($\vec p^2/2m\rightarrow$ 3 degrees of freedom), the rotation ($\vec L^2/2I\rightarrow$ 2 degrees of freedom for the 2 Euler angles) and the vibration (which also accounts for 2 degrees of freedom), in total, this gives:
$$C_v = \frac{3}{2}kT + \frac{2}{2}kT + \frac{2}{2}kT = \frac{7}{2}kT$$
In reality, we measured $C_v\simeq\frac{5}{2}kT$. Smart people like Jeans tried to fix this by modifying the equipartition (he argued that some degrees of freedom would take a large time to equilibrate, like the rotation ones or the vibration ones. Implying that the equipartition wouldn't hold for a "finite time" for every degrees of freedom and that we could omit some of them...).
But the real fix was found by Einstein in his 1907 paper. He showed that  "quantising" the energy of the atoms in a crystal would result in the correct heat capacity (forgetting Fermi, Debye and others which built upon his theory, remarquably, Einstein gave at the end of his paper the issues that had to be solved in order to get a better result, and they were exactly the ones fixed by his contemporary). Basically the vibrating degree of freedom is frozen.  The solid, at ambient temperature doesn't have enough energy to excite the first vibrating energy level. This means that we can, kind off, omit it in the calculation of the energy. Off course, this is an hand waving argument, the real thing is the fully quantum description.
A: The gold standard in physics is experiment.  A theory is only useful if it produces results that match experiment within desired limits.
We use quantum mechanics because it produces results which match those of experiment.
Classical mechanics does not reproduce those results so we do not use it.  It predicts qualitative and quantitative results which are simply wrong.  A trivial example is that classical theories do not model the molecular bonding of atoms (or even single atoms) satisfactorily.  It's not that you cannot use classical mechanics at all, it's that classical mechanics fails too often and the models are often very contrived and not consistent.  Quantum theory "covers all the bases".
A: It all started classical. Drude made his theory of conductivity, and showed that $\sigma = n e^2\tau/m$, $\tau$ being "characteristic scattering time". What is it? Time is not an intrinsic property of a material. The more proper quantity is the mean free length $\lambda$ which can be thought of as typical distance between scattering centers; then $\tau = \lambda/v$, $v$ being the speed of electrons. But what is it? Drude was thinking of electrons as a sort of classical gas for which we all know how to calculate a typical $v_{typical} \propto \sqrt{T}$. But such temperature behavior was in clear contradiction with what we observe in metals. The worst thing is that even the absolute value is orders of magnitude off. The typical speeds of electrons in metals at room temperatures (as calculated from conductivity) correspond to tens of thousands of Kelvins.
Pauli has solved this riddle by applying his exclusion principle for metals and everything made perfect sense. Landau came later to show how Pauli principle makes sure you can treat electrons as a gas of almost free electrons despite humongous Coulomb repulsion between them. So you see, you have to use QM to have at least crudely sensible understanding of them.
