Hydrogen Atom Radial Functions
Hydrogen Atom Radial Functions
The eigenfunctions in spherical coordinates for the hydrogen atom are (r,θ,φ)=(r)(θ,φ), where (r) and (θ,φ) are the solutions to the radial and angular parts of the Schrödinger equation, respectively, and , , and are the principal, orbital, and magnetic quantum numbers with allowed values , and . The (θ,φ) are the spherical harmonics and the radial functions are (r)=(2r/n), where (x) is the order associated Laguerre polynomial and is the Bohr radius. The left graphic shows the radial probability density (r) and the expectation value ≡r=1+1, and the right graphic shows the radial function.
ϕ
n,ℓ.m
R
n,ℓ
m
Y
ℓ
R
n,ℓ
m
Y
ℓ
n
ℓ
m
n=0,1,2,…,n1,
ℓ=0,1,2,…,n1
m=0,±1,±2,…,±ℓ
m
Y
ℓ
R
n,ℓ
(n1ℓ)!
2
3
n[(n+ℓ)!]
ℓ+3/2
2
n
a
0
ℓ
r
r/n
a
0
e
2ℓ+1
L
n+ℓ
a
0
a
L
p
th
p
a
0
2
r
R
n,ℓ
2

〈r〉
n,ℓ
ϕ
n,ℓ.m
ϕ
n,ℓ.m
2
n
a
0
1
2
ℓ(ℓ+1)
2
n