Do electrons in the conduction band have a quantum number? Sorry if this is a silly question, but I haven't seen the concept anywhere and was wondering why.
If valence electrons can have a quantum number, why can't conduction electrons?
Is there something that makes this idea impractical?
 A: You can say they have one, but it doesn't mean very much.
First off, as a technicality that will matter a bit: electrons don't "have" quantum numbers, electrons are described using quantum numbers.  We find the quantum numbers do a good job of describing the behaviors of electrons, but the electrons themselves simply do whatever electrons do.
In theory you could say a conduction band electron is "bound" to the entire cluster of atoms.  However, the numbers get a bit squirly.  An electron is described using 3 quantum numbers:


*

*n - Principle quantum number, describing the "size" of the orbital.  Because a conduction band can have a lot of electrons, clearly this is going to be rather large n.

*l - Angular quantum number.  This accounts for the many different shapes of orbitals.  it ranges from 0 to n-1

*m - Magnetic quantum number.  This accounts for the orientation of the different orbitals.  Its values range from -l to l.


If you consider a high n value, such as you would need to meaningfully assign quantum numbers to a hodge podge of conduction electrons, you'll find that the number of valid quantum numbers is O(n^3), which is a mathematically precise way of saying it grows by the cubic power of n.
The primary value of analyzing systems with quantum numbers is the ability to use the Pauli Exclusion Principle to make statements about what can and cannot occur.  When n is small, this is a big deal because the orbitals can become relatively full.  However, for larger n, there are so many more potential quantum numbers than actual electrons that the Exclusion Principle ceases to provide very much useful information.  It still holds true, it just isn't quite as useful as it was for small n.
A: Every quantum state is described by some list of quantum numbers. But for most states, those quantum numbers have no relation to the quantum numbers in an atom. For example, the quantum numbers for an electron in a infinite square well can't be naturally stated in terms of atomic orbitals. So if I reformulate your question as "Do electrons in a band material have a quantum number similar to the ones in a single atom?," then it is a good and sensible question. And it turns out that the answer is yes, there is indeed a relationship between the quantum numbers of a single atom and the band structure of a material.
In a simple tight binding model (the wiki page doesn't look like a great introduction, unfortunately, but maybe it has a link that is better), one starts with separate atoms with electronic orbitals and asks how the energy levels change as the atoms are brought together to form a uniform material. And it turns out that the atomic orbitals like 1s, 2s, 2p, and so on go from being single energy levels to broad bands. So, if you start with a bunch of atoms that individually have electrons filling $n$ energy levels, the material formed from a collection of this atom will have $n$ bands, although these bands could be partially or totally degenerate depending on the specifics of the interactions between atoms. Each electron, instead of having a quantum number for its orbital, now has a similar number that says which band it is in. It also has a second set of quantum numbers (three if you are in 3D), the quasimomentum quantum numbers, which say exactly where in the band the electron is. Unlike the band number, these don't correspond to anything in the original atoms.
In some very simple cases, the mapping from orbital to band is exact, so that you might be able to say that an atom is in the "1s band." However, in general that is not true. For example, the first two bands might be different combinations of what were the 1s and 2s orbitals. So although you always get as many bands as you had orbitals, the original orbital labels won't always be the same labels that apply to the bands.
A: A free electron does not have a quantum number. For the hydrogen atom, if you provide 13.6eV of energy, the electron is free- it is no longer 'bound' to the atom, thus it's energy is no longer quantized. Atoms do not form bands. Atoms have discrete energy levels, there is no such thing as valence and conduction bands for an atom.
If you bring N atoms close to each other, like in a semiconductor crystal, you will form N energy levels, N/2 are bonding orbitals (valence band) and N/2 are anti-bonding orbitals of higher energy, and they form the conduction bands. The difference between the highest valence band state and lowest conduction band state is the bandgap.
The conduction band states in a crystal are quantized, but they are so close together (if N is large) that the discrete levels are smeared into bands.
I believe your confusion arose from not quite knowing the difference between the energy bands in atoms vs bulk solids (a collection of atoms).
