Hydrogen Atom Radial Functions

quantum numbers n

1

2

3

4

l

0

The eigenfunctions in spherical coordinates for the hydrogen atom are

ϕ

n,ℓ.m

(r,θ,φ)=

R

n,ℓ

(r)

m

Y

ℓ

(θ,φ)

, where

R

n,ℓ

(r)

and

m

Y

ℓ

(θ,φ)

are the solutions to the radial and angular parts of the Schrödinger equation, respectively, and

n

,

ℓ

, and

m

are the principal, orbital, and magnetic quantum numbers with allowed values

n=0,1,2,…,n-1,

ℓ=0,1,2,…,n-1

, and

m=0,±1,±2,…,±ℓ

. The

m

Y

ℓ

(θ,φ)

are the spherical harmonics and the radial functions are

R

n,ℓ

(r)=

(n-1-ℓ)!

2

3

n[(n+ℓ)!]

ℓ+3/2

2

n

a

0

ℓ

r

-r/n

a

0

e

2ℓ+1

L

n+ℓ

(2r/n

a

0

)

, where

a

L

p

(x)

is the

th

p

-order associated Laguerre polynomial and

a

0

is the Bohr radius. The left graphic shows the radial probability density

2

r



R

n,ℓ

(r)

2

|

and the expectation value

〈r〉

n,ℓ

≡

ϕ

n,ℓ.m

r

ϕ

n,ℓ.m

=

2

n

a

0

1+

1

2

1-

ℓ(ℓ+1)

2

n



, and the right graphic shows the radial function.