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Electron Probability Distribution for the Hydrogen Atom

electron
quantum
numbers
n
1
2
3
l
0
1
m
-1
0
1
electron
probability
density (au)
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 13, along with the allowed values of the quantum numbers
l
and
m
.
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