WOLFRAM|DEMONSTRATIONS PROJECT

Quantum Alchemy

​
show atomic orbital
n
1
2
3
4
l
0
1
2
show all orbitals
Schrödinger [1, 2] made use of a factorization method on the hydrogen atom radial equation to show that all solutions can be generated starting with the ground state. Such procedures are now usually categorized as supersymmetric quantum mechanics. In an earlier publication [3], we dubbed this modern form of alchemy: "quantum alchemy".
The Schrödinger equation for a nonrelativistic hydrogenic atom has the form
-
1
2
2
∇
-
Z
r
ψ=Eψ
, with use of atomic units and infinite nuclear mass (
ℏ=
m
e
=e=1
,
m
p
=∞
). In our specific examples we take
Z=1
for the hydrogen atom itself. The solution is separable in spherical polar coordinates:
ψ
n l m
(r,θ,ϕ)=
R
n l
(r)
Y
l m
(θ,ϕ)
. The
Y
l m
(θ,ϕ)
are spherical harmonics; their transformation properties are well documented and we need not consider them further. We use the value
m=0
in our illustrations of atomic orbitals. For bound states with
E
n
=-
2
Z
2
2
n
, the normalized radial function can be expressed
R
n l
(ρ)=
(n-1-l)!
2
3
n[(n+l)!]
3/2
2
n
l
ρ
-ρ/2
e
2l+1
L
n-l-1
(ρ)
, where
L
is an associated Laguerre polynomial and
ρ=
2Zr
n
.
It is simpler to work with the reduced radial function
P
n l
(r)=r
R
n l
(r)
, which obeys the pseudo one-dimensional differential equation
-
1
2
2
d
d 
2
r
+
l(l+1)
2
2
r
-
Z
r
P
n l
(r)=
E
n
P
n l
(r)
. We consider the two "alchemical" transformations

l
and

n l
, which have the following actions:

l+1
P
n,l
​
(r)=const
P
n,l+1
(r)
and

n l
P
n,l
(r)=const
P
n+1,l
(r)
. The first operator, for example, turns a
2s
-orbital into a
2p
-orbital, while the second turns it into a
3s
-orbital.
This Demonstration shows a sequence of steps that can convert the ground state
1s
orbital into a chosen orbital with user specified values of
n
and
l
, up to
n=4
and
l=3
. Plots of the radial functions
P
10
(r)
and
P
n l
(r)
are also shown.