# Electron Probability Distribution for the Hydrogen Atom

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 can be solved by separation of variables in the form . The radial and angular components are Laguerre and Legendre functions, thus (r)∝ and (θ,ϕ)∝(cosθ), respectively. Here, =5.29Å is the first Bohr radius, and are the integers in the ranges (principal quantum number), (angular momentum quantum number), and (magnetic quantum number). The probability of finding an electron at a specific location is given by , where is the normalization constant, such that .

(r,θ,ϕ)

ψ(r,θ,ϕ)=(r)(θ,ϕ)

R

nl

Y

lm

R

nl

r(n)

a

0

e

l

2r

n

a

0

2l+1

L

n-l-1

2r

n

a

0

Y

lm

m

P

l

imϕ

e

a

0

n,l,m

n=1,2,⋯

l=0,⋯,n-1

m=-l,-l+1,⋯,0,1,⋯,l

P(r,θ,ϕ)=Nψ

*

ψ

N

∫P(r,θ,ϕ)sinθdrdθdϕ=1

2

r

This Demonstration shows the three-dimensional distribution of probability using color when are specified. Only the real part of is considered. Putting the proton at the origin, the probability density is displayed within a sphere of radius . You can select the principal quantum number in the range 1–3, along with the allowed values of the quantum numbers and .

P(r,θ,ϕ)

n,l,m

ψ

15

a

0

n

l

m