The Harmonic Oscillator

H = -(ħ²/(2m))(∂²/∂x²) + ½mω²x² is the harmonic oscillator Hamiltonian.
(∂²/∂x²)Φ(x) + (2m/ħ²)(E - ½mω²x²)Φ(x) = 0 is the eigenvalue equation.
The eigenvalues are E_n = (n + ½)ħω, n = 0, 1, 2, ... .

The normalized ground state wave function of the 1-dimensional harmonic oscillator is
Φ₀(x) = (mω/(πħ))^¼exp(-½mωx²/ħ).
The wave functions of the excited states are
Φ_n(x) = (n! 2ⁿ)^-½(β/√π)^½H_n(η) exp(-½η²),
where η = (mω/ħ)^½ x = βx.

Specifically,
Φ₁(x) = ((4/π)(mω/ħ)³)^¼ x exp(-½mωx²/ħ),
Φ₂(x) = (mω/(4πħ))^¼[2mωx²/ħ - 1] exp(-½mωx²/ħ).
Φ_n(x) is the product of exp(-½mωx²/ħ) and a polynomial of degree n and parity (-1)ⁿ called a Hermite polynomial.

Defining scaled operators X_s = (mω/ħ)^½X,  P_s =  (mωħ)^-½P, and new operators
a = (2)^-½(X_s + iP_s), and its adjoint, a^† = (2)^-½(X_s - iP_s), we can write
H = ħω(a^†a + ½) = ħω(aa^† - ½) = ½ħω(aa^† + a^†a).
The operators a and a^† do not commute.
[a, a^†] = 1.
H and a^†a  have the same eigenstates {|n>}.
a^†a|n> = n|n>, H|n> = (n + ½)ħω|n>.
a|n> = √(n) |n-1>,  a^†|n> = √(n+1) |n+1>.

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, L_i] = 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 L_z.  We denote these basis states |k,l,m> and the corresponding eigenfunctions by ψ_klm(r,θ,φ).  We have

H|klm> = E_kl|klm>,  L²|klm> = ħ²l(l + 1)|klm>,  L_z|klm> = mħ|klm>.

The wave function ψ_klm(r,θ,φ) = R_kl(r)Y_lm(θ,φ) = [u_kl(r)/r]Y_lm(θ,φ) is a product of a radial function R_kl(r) and the spherical harmonic Y_lm(θ,φ).  The differential equation for u_kl(r) is
[-(ħ²/(2m))(∂²/∂r²) + ħ²l(l + 1)/(2mr²) + U(r)]u_kl(r) = E_klu_kl(r).

The asymptotic behavior of R_kl(r)
Near the origin the radial behavior of an acceptable wave function of a particle moving in a central potential is proportional to r^l, (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))∇²_rΦ(r) + U(r)Φ(r) = E_rΦ(r).
With Φ_klm(r) = R_kl(r)Y_lm(θ,φ) = [u_kl(r)/r]Y_lm(θ,φ)  we have
[-(ħ²/(2μ))(∂²/∂r²) + ħ²l(l + 1)/(2μr²) + U(r)]u_kl(r) = E_klu_kl(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 = m_em_p/(m_e + m_p) ≈ m_e.
Writing
Φ_nlm(r) = R_nl(r)Y_lm(θ,φ) = [u_nl(r)/r]Y_lm(θ,φ)
we find
u₁₀(r) = 2a₀^-3/2 r exp(-r/a₀),  Φ₁₀₀(r) = (πa₀³)^-½ exp(-r/a₀).
Here a₀ = ħ²/(μe²).
The ground state energy of the hydrogen atom is -E_I. 
E_I = 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.
E_n = -E_I/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 E_n = -μe⁴/(2ħ²n²) = -hcR_H/n²
R_H is the Rydberg constant.
E_n' - E_n = hcR_H(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 E_I by E_I' = μ'Z²e⁴/(2ħ²) = E_I(μ'/μ)Z².