Why are quantum numbers assigned to electrons? Specifically the principal quantum number ($n$), orbital quantum number ($l$) and orbital magnetic quantum number ($m_{l}$). For systems like the hydrogen atom, these quantum numbers arise from the Schrödinger equation which involves:

*

*A potential energy function, for the system of the electron and the nucleus

*A wave function, also characteristic of the same system, determined by the electron and it's interaction with it's environment

Wouldn't it then be more appropriate to assign quantum numbers to the electron-nucleus system rather than the electron itself?
Several sources always describe the quantum numbers (particularly in multi-electron systems) to be a characteristic of an electron, including the exclusion principle, so I am unsure if my reasoning is correct.
Here's the paragraph from the Wikipedia page on the exclusion principle:

it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers: n, the principal quantum number; ℓ, the azimuthal quantum number; $m_ℓ$, the magnetic quantum number; and $m_s$, the spin quantum number.

 A: Yes, these numbers are assigned to the electron-nucleus system. Usually (as in classical mechanics) the treatment of a hydrogen atom starts with separating the motion of the center-of-mass of the atom and the relative motion of the electron and the proton, reducing a two-body problem to a one-body problem in an effective potential (and with an effective mass). This is easily overlooked in the quantum forest, but is actually found in many basic QM books.
It is for the wave function of this relative motion that one defines the quantum numbers mentioned in the OP. The reason why one often speaks of electrons making transitions and so on, is because proton is a thousand times heavier than the electron, and consequently the center of mass pretty much coincides with that of the proton, while the internal motion of the atom is pretty much that of the electron.
A: We Specify quantum numbers to electron because they are extremely useful in locating the electron. You may ask how?
The answer is these variable are very important to solve schrodinger wave equation that tells you about the wavefunction Ψ. Ψ doesn't have a physical significance but it's square tells us about the probability density of the electron(where there is higher probability of finding a electron).
