Consider a spinless particle of mass m in a central potential U(r). The
Hamiltonian operator is

-(ħ^{2}/(2m))(1/r)(∂^{2}/∂r^{2})r
+ L^{2}/(2mr^{2}) + U(r),

[H,L_{i}] = 0, [H,L^{2}] = 0.

The angular momentum **L** of the particle is a constant of motion. We
can find a common eigenbasis of H, L^{2} 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>, L^{2}|klm> =
ħ^{2}l(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

[-(ħ^{2}/(2m))(∂^{2}/∂r^{2})
+ ħ^{2}l(l + 1)/(2mr^{2})
+ U(r)]u_{kl}(r) = E_{kl}u_{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

Two interacting particles

-(ħ

With Φ

[-(ħ

Here μ is the reduced mass.

The time-independent Schroedinger equation for the hydrogen atom is

H(**r**,**p**)Φ(**r**) = [-(ħ^{2}/(2μ))∇^{2}
- e^{2}/r]Φ(**r**) = EΦ(**r**),

where μ = m_{e}m_{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

u_{10}(r) = 2a_{0}^{-3/2 }r exp(-r/a_{0}),
Φ_{100}(r) = (πa_{0}^{3})^{-½
}exp(-r/a_{0}).

Here a_{0} = ħ^{2}/(μe^{2}).

The ground state energy of the hydrogen atom is -E_{I}.

E_{I} = e^{2}/(2a_{0}) = μe^{4}/(2ħ^{2})
= (μ/2)α^{2}c^{2} = 13.6 eV,

where α^{2}
= e^{4}/(ħ^{2}c^{2}),
α = e^{2}/(ħc)
= 1/137. α is the fine structure constant.

E_{n} = -E_{I}/n^{2} = -(μ/(2n^{2}))α^{2}c^{2}
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} = -μe^{4}/(2ħ^{2}n^{2})
= -hcR_{H}/n^{2}

R_{H} is the Rydberg constant.

E_{n'} - E_{n}
= hcR_{H}(1/n^{2} -
1/n'^{2}).

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 a_{0}
by a_{0}' = ħ^{2}/(μ'Ze^{2})
= a_{0}(μ/μ')(1/Z),
and in the eigenvalues of the Hamiltonian of the hydrogen atom we replace E_{I}
by E_{I}' = μ'Z^{2}e^{4}/(2ħ^{2})
= E_{I}(μ'/μ)Z^{2}.