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WOLFRAM|DEMONSTRATIONS PROJECT

Particle in an Infinite Spherical Well

​
display
wavefunction
energy levels
quantum numbers:
n
1
2
3
l
0
1
2
m
0
1
A particle of mass
M
in an infinite spherical potential well of radius
R
is described by the Schrödinger equation
-
2
ℏ
2M
2
∇
ψ=Eψ
. The wavefunction is separable in spherical polar coordinates, such that
ψ
nlm
(r,θ,ϕ)=
N
nlm
j
l
(
k
ln
r)
Y
lm
(θ,ϕ)
, where
Y
lm
is a spherical harmonic,
j
l
a spherical Bessel function, and
N
is a normalization constant. The boundary condition that
ψ=0
at
r=R
is fulfilled when
k
ln
R
is the
th
n
zero of the spherical Bessel function
j
l
. The quantized energy levels are then given by
E
nl
=
2
ℏ
2
k
ln
2M
2
R
and are
(2l+1)
-fold degenerate with
m=0,±1,±2,…,±l
. The conventional code is used to label angular momentum states, with
s,p,d,f,…
representing
l=0,1,2,3,…
. Unlike atomic orbitals, the
l
-values are not limited by
n
; thus one will encounter states designated
1p,1d,2f
, etc.
This Demonstration shows contour plots on a cross section of the sphere for the lower-energy eigenfunctions with
n=1,2,3
and
l=0,1,2
. For
m>0
, the eigenfunctions are complex. In all cases, the real parts of
ψ
nlm
(r,θ,ϕ)
are drawn. The wavefunction is positive in the blue regions and negative in the white regions. You can also view an energy-level diagram, with each dash representing the degenerate set of eigenstates for given
l
.
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