WOLFRAM|DEMONSTRATIONS PROJECT

Connection between Quantum-Mechanical Hydrogen Atom and Harmonic Oscillator

​
quantum numbers:
n
1
2
3
4
ℓ
0
1
plots
wavefunctions
radial distribution functions
The bound states of the hydrogen atom are governed by the geometrical symmetry
SO(3)
(not considering the full dynamical symmetry
SO(4)
). Similarly, the two-dimensional isotropic harmonic oscillator exhibits the symmetry
SU(2)
. To anyone versed in the theory of Lie groups, it would not be surprising that there might be an explicit connection between these two problems, in view of the local isomorphism between the corresponding Lie algebras
so(3)
and
su(2)
.
The radial Schrödinger equation for a hydrogen-like system, in atomic units, is given by
-
1
2
2
r
d
dr
2
r
dR
dr
+
l(l+1)
2
2
r
R(r)-
Z
r
R(r)=-
2
Z
2
2
n
R(r),l=0,1,2,…,n=l+1,l+2,…
,
with the unnormalized solutions
R
nl
(r)=
l
r
-Zr/n
e
2l+1
L
n-l-1
2Zr
n
,
where
L
is an associated Laguerre polynomial. Consider now a two-dimensional isotropic harmonic oscillator, expressed in polar coordinates. The associated radial Schrödinger equation takes the form
-
1
2ρ

ρ
ρ
dP
dρ
+
2
m
2
2
ρ
P(ρ)+
1
2
2
ω
2
ρ
P(ρ)=
n
osc
ωP(ρ),m=0,±1,±2,…,
n
osc
=|m|+1,|m|+3,|m|+5,…
Using DSolve, we find the unnormalized solutions:
P
m
n
osc
(ρ)=
m
ρ
-ω
2
ρ
2
e
m
L

n
osc
-m-12
ω
2
ρ

.
The solutions of the two problems can be made equivalent by the substitutions:
r
nω
2Z
2
ρ
,
m2l+1
,
n
osc
2n
.
For selected values of
n
and
l
, the graphic shows plots of the radial wavefunctions
R(r)
and
P(ρ)
, as well as the corresponding radial distribution functions (RDFs)
2
r
2
R(r)
and
ρ
2
P(ρ)
.