Electron Probability Distribution for the Hydrogen Atom

The electron in a hydrogen atom is described by the Schrödinger equation. The time-independent Schrödinger equation in spherical polar coordinates

(r,θ,ϕ)

can be solved by separation of variables in the form

ψ(r,θ,ϕ)=

R

nl

(r)

Y

lm

(θ,ϕ)

. The radial and angular components are Laguerre and Legendre functions, thus

R

nl

(r)∝

r(n

a

0

)

e

l

2r

n

a

0

2l+1

L

n-l-1

2r

n

a

0

and

Y

lm

(θ,ϕ)∝

m

P

l

(cosθ)

imϕ

e

, respectively. Here,

a

0

=5.29Å

is the first Bohr radius, and

n,l,m

are the integers in the ranges

n=1,2,⋯

(principal quantum number),

l=0,⋯,n-1

(angular momentum quantum number), and

m=-l,-l+1,⋯,0,1,⋯,l

(magnetic quantum number). The probability of finding an electron at a specific location is given by

P(r,θ,ϕ)=Nψ

*

ψ

, where

N

is the normalization constant, such that

∫P(r,θ,ϕ)

2

r

sinθdrdθdϕ=1

.

This Demonstration shows the three-dimensional distribution of probability

P(r,θ,ϕ)

using color when

n,l,m

are specified. Only the real part of

ψ

is considered. Putting the proton at the origin, the probability density is displayed within a sphere of radius

15

a

0

. You can select the principal quantum number

n

in the range 1–3, along with the allowed values of the quantum numbers

l

and

m

.