## 3D eigenvalue problems

### A particle in a central potential

Consider a spinless particle of mass m in a central potential U(r).  The Hamiltonian operator is
-(ħ2/(2m))(1/r)(∂2/∂r2)r + L2/(2mr2) + U(r),
[H,Li] = 0, [H,L2] = 0.
The angular momentum L of the particle is a constant of motion.  We can find a common eigenbasis of H, L2 and Lz.  We denote these basis states |k,l,m> and the corresponding eigenfunctions by ψklm(r,θ,φ).  We have

H|klm> = Ekl|klm>,  L2|klm> = ħ2l(l + 1)|klm>,  Lz|klm> = mħ|klm>.

The wave function ψklm(r,θ,φ) = Rkl(r)Ylm(θ,φ) = [ukl(r)/r]Ylm(θ,φ) is a product of a radial function Rkl(r) and the spherical harmonic Ylm(θ,φ).  The differential equation for ukl(r) is
[-(ħ2/(2m))(∂2/∂r2) + ħ2l(l + 1)/(2mr2) + U(r)]ukl(r) = Eklukl(r).

The asymptotic behavior of Rkl(r)
Near the origin the radial behavior of an acceptable wave function of a particle moving in a central potential is proportional to rl, (if  |U(r)| < |C/r2| as r --> 0).

Two interacting particles
We are often only interested in the relative motion.  If the mutual interaction depends only on the distance between the particles r = |r1-r2|, then the eigenvalue equation for the relative motion becomes
-(ħ2/(2μ))∇2rΦ(r) + U(r)Φ(r) = ErΦ(r).
With Φklm(r) = Rkl(r)Ylm(θ,φ) = [ukl(r)/r]Ylm(θ,φ)  we have
[-(ħ2/(2μ))(∂2/∂r2) + ħ2l(l + 1)/(2μr2) + U(r)]ukl(r) = Eklukl(r).
Here μ is the reduced mass.

### The hydrogen atom

The time-independent Schroedinger equation for the hydrogen atom is
H(r,p)Φ(r) = [-(ħ2/(2μ))∇2 - e2/r]Φ(r) = EΦ(r),
where μ = memp/(me + mp) ≈ me.
Writing
Φnlm(r) = Rnl(r)Ylm(θ,φ) = [unl(r)/r]Ylm(θ,φ)
we find
u10(r) = 2a0-3/2 r exp(-r/a0),  Φ100(r) = (πa03)exp(-r/a0).
Here a0 = ħ2/(μe2).
The ground state energy of the hydrogen atom is -EI
EI = e2/(2a0) = μe4/(2ħ2) = (μ/2)α2c2 = 13.6 eV,
where α2 = e4/(ħ2c2),  α = e2/(ħc) = 1/137.  α is the fine structure constant.
En = -EI/n2 = -(μ/(2n2))α2c2 is the energy of the nth excited state.  Here n is called the principal quantum number, n fixes the energy of the eigenstate.  Given n, l can take on n possible values  l = 0, 1, 2, ..., (n - 1).  n characterizes an electron shell, which contains n subshells characterized by l.  Each subshell contains 2l + 1 distinct states.
We also write En = -μe4/(2ħ2n2) = -hcRH/n2
RH is the Rydberg constant.
En' - En = hcRH(1/n2 - 1/n'2).

### Hydrogenic atoms

To find the eigenfunctions and eigenvalues of the Hamiltonian of a hydrogenic atom we replace in the eigenfunctions of the Hamiltonian of the hydrogen atom a0 by a0' = ħ2/(μ'Ze2) = a0(μ/μ')(1/Z), and in the eigenvalues of the Hamiltonian of the hydrogen atom we replace EI by EI' = μ'Z2e4/(2ħ2) = EI(μ'/μ)Z2.