Hydrogen atom- Eigenvalue/function relation I have been given the following Question:

The energy eigenstates of the atomic electron are usually described by wave
functions $ψ_{nℓm}(r)$.
Relate each of $n, ℓ,$ and $m$ to the eigenvalue of a specific operator by giving
the eigenvalue equation for this operator acting on $ψ_{nℓm}(r)$.


I understand the following eigenvalue/function relations:
$$\hat{\vec{L}^2} Y_{ℓm}=\hbar^2 ℓ(ℓ+1) Y_{ℓm}.$$
and;
$$\hat{L_z} Y_{ℓm}=\hbar m Y_{ℓm}.$$
But I don't understand where the principle quantum number, $n$ comes into things. If someone could explain, that'd be great. Thanks.
 A: n is for the energy of the electron, the eigenvalue of the Hamiltonian. It is called principal, because it should be the basic being related to energy.
It is n because it is natural, in the case of H, it is En=-13.6eV/n^2.
$$\sum_{\ell=0}^{\ell=n-1}(2\ell + 1)= n^2$$
It was first used with Bohr H atom, he used n for the quantization of angular momentum, as n the allowed orbit.
$L = n{h \over 2\pi} = n\hbar$
But the n you are talking about is the solution of the Schrodinger equation for H. This n that you get by solving the Schrodinger equation is the allowed energy state.

Hamilton's equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states { | n ⟩ } , which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,

⟨ n ′ | n ⟩ = δ n n ′ 


Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.
The instantaneous state of the system at time t, | ψ ( t ) ⟩  , can be expanded in terms of these basis states:

| ψ ( t ) ⟩ = ∑ n a n ( t ) | n ⟩  


where

a n ( t ) = ⟨ n | ψ ( t ) ⟩ . 

Please look at Hamilton's equations in https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)
