WOLFRAM|DEMONSTRATIONS PROJECT

Hydrogenic Radial Functions via Supersymmetry

​
n
1
2
3
4
l
0
operator
A
l
+
A
l-1
formulas
plots
energy levels
An application of supersymmetric quantum mechanics enables all the bound-state radial functions for the hydrogen atom to be evaluated using first-order differential operators, without any explicit reference to Laguerre polynomials.
The nonrelativistic hydrogen-like system with atomic number
Z
and assumed infinite nuclear mass satisfies the Schrödinger equation
-
1
2
2
∇
-
Z
r
+
2
Z
2
2
n
ψ
nlm
(r,θ,ϕ)=0
, in atomic units
ℏ=m=e=1
. Separation of variables in spherical polar coordinates gives
ψ
nlm
(r,θ,ϕ)=
R
nl
(r)
Y
lm
(θ,ϕ)
. Defining the reduced radial function
P(r)=rR(r)
, the radial equation can be expressed as
-
2
d
d
2
r
+
l(l+1)
2
r
-
2Z
r
+
2
Z
2
n
P
nl
(r)=0
. For the case
l=n-1
(the 1
s
, 2
p
, 3
d
, 4
f
, … states) the radial function has the nodeless form
P
n,n-1
(r)=const
n
r
-Zr/n
e
.
As shown in the Details section, operators
A
l+1
and
+
A
l
are defined with the effect of lowering or raising the quantum number
l
by 1, respectively, when applied to the radial function
P
nl
(r)
, namely,
A
l+1
P
n,l
(r)=const
P
n,l+1
(r)
and
+
A
l
P
nl
(r)=const
P
n,l-1
(r)
. The constants are most easily determined after the fact by the normalization conditions
∞
∫
0
2
P
nl
(r)
dr=1
.
In this Demonstration, you can plot any radial function with
n=1
to
4
and show the result of applying
A
l
(red curve) or
+
A
l
(blue curve). The plots pertain to the case
Z=1
. You can also choose to view the results as formulas (with variable
Z
) or on an energy-level diagram.