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We know that good quantum numbers are associated with operators that commute with the Hamiltonian of the system.

For example consider an Hydrogen atom without spin, we know that the good quantum numbers are n,l and m, which means that angular momentum is conserved.

How about in a solid? Because of the periodicity of wave vector k to respect the first Brillouin zone, my guess is that k is a good quantum number in this case. A good way to confirm this is that the cristalline linear momentum is conserved in crystals, not the "normal" linear momentum.

But is this the only quantum number?

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As you already said it yourself:

We know that good quantum numbers are associated with operators that commute with the Hamiltonian of the system.

This is correct.

I assume that you already know that we say that Hamiltonian $H$ has a symmetry represented by the unitary or antiunitary operator $U$ if: $$ [H,U]=0 $$ That means that your question is equivalent to the question: "What are the symmetries of our system?"

As you correctly pointed out, the Hydrogen atom has rotational SO(3) symmetry. The generators of SO(3) group are the angular momentum operator (which do not commute between themselves but all of them commute with $L^2$), so we can conclude that: \begin{eqnarray} [H,L_z]&=&0 \quad \text{from this equation you see that quantum number m is good}\\ [H,L^2]&=&0\quad \text{from this equation you see that quantum number l is good} \end{eqnarray}

Similarly in crystal you know that you have translational symmetry, so you can conclude that: $$ [H,T_a]=0\quad $$ where $T_a$ is the generator of translations . From this equation you obtain that crystal momentum k is good quantum number.

But is this the only quantum number?

As I already pointed out, that depends on whether there exist additional symmetries in your system (have in mind that the generators of those symmetries need to commute not only with the Hamiltonian $H$, but also with all the other generators that you picked). That said, you certainly can have other quantum numbers, but we need to specify the system in order to find which other good quantum numbers are present. For example, if our system has inversion symmetry, then parity is also a good quantum number.

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  • $\begingroup$ Thanks for your answer. Do you have any references about this topic? $\endgroup$ – Lucas Lopes Apr 1 '20 at 17:35
  • $\begingroup$ Well, as I said, your question is basically about symmetry. So any group theory with applications in physics will do just fine. I liked "Group Theory and Its Application to Physical Problems" by Morton Hamermesh. $\endgroup$ – RedGiant Apr 1 '20 at 21:28

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