Three-Dimensional Isotropic Harmonic Oscillator

​
display
contours
R
n,l
(r)
E
n,l
ω
0.1
quantum numbers:
l
0
1
2
3
n
0
2
4
6
m
0
The isotropic three-dimensional harmonic oscillator is described by the Schrödinger equation
-
1
2
2
∇
ψ+
1
2
2
ω
2
r
ψ=Eψ
, in units such that
ℏ=m=1
. The wavefunction is separable in Cartesian coordinates, giving a product of three one-dimensional oscillators with total energies
E
n
1
n
2
n
3
=
n
1
+
n
2
+
n
3
+
3
2
ℏω
. More interesting is the solution separable in spherical polar coordinates:
ψ
nlm
(r,θ,ϕ)=
R
nl
(r)
Y
lm
(θ,ϕ)
, with the radial function
R
nl
(r=)
N
nl
l
r
-ω
2
r
/2
e
​
l+1/2
L
(n-l}/2
(ω
2
r
)
. Here,
L
is an associated Laguerre polynomial,
Y
​
, a spherical harmonic and
N
, a normalization constant. The energy levels are then given by
E
n
=n+
3
2
ℏω
, being
1
2
(n+1)(n+2)
-fold degenerate. For a given angular momentum quantum number
l
, the possible values of
n
are
l,l+2,l+4,…
. The conventional code is used to label angular momentum states, with
s,p,d,f,…
representing
l=0,1,2,3,…
.
This Demonstration shows contour plots in the
x-z
plane for the lower-energy eigenfunctions with
l=0
to
3
. For
m>0
, the eigenfunctions are complex. In all cases, the real parts of
ψ
nlm
(r,θ,ϕ)
are drawn. The wavefunctions are positive in the blue regions and negative in the white regions. The radial functions are also plotted, as well as an energy-level diagram, with each dash representing the degenerate set of
2l+1
eigenstates for a given
l
.
The pattern of degeneracies for a three-dimensional oscillator implies invariance under an SU(3) Lie algebra, the same as the gauge group describing the color symmetry of strong interactions.

Details

Reference: Wikipedia article Quantum Harmonic Oscillator

External Links

Schrödinger Equation (Wolfram MathWorld)
Laguerre Polynomial (Wolfram MathWorld)

Permanent Citation

S. M. Blinder
​
​"Three-Dimensional Isotropic Harmonic Oscillator"​
​http://demonstrations.wolfram.com/ThreeDimensionalIsotropicHarmonicOscillator/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011