WOLFRAM|DEMONSTRATIONS PROJECT

The Hydrogen Atom in Parabolic Coordinates

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quantum numbers:
n
1
0
1
2
3
4
n
2
0
1
2
3
4
|m|
0
1
2
3
4
electric field:
ℰ
0
The Schrödinger equation for the hydrogen atom,
-
1
2
2
∇
ψ(r)-
1
r
ψ(r)=Eψ(r)
(in atomic units
ℏ=m=e=1
), can be separated and solved in parabolic coordinates
(ξ,η,ϕ)
as well as in the more conventional spherical polar coordinates
(r,θ,ϕ)
. This is an indication of degeneracy in higher eigenstates and is connected to the existence of a "hidden symmetry", namely the
SO(4)
Lie algebra associated with the Coulomb problem. Parabolic coordinates can be defined by
ξ=r(1+cosθ)=r+z
,
η=r(1-cosθ)=r-z
, with the same
ϕ=arctan(y/x)
as in spherical coordinates. The wavefunction is separable in the form
ψ
n
1
n
2
m
(ξ,η,ϕ)=N
f
1
(ξ)
f
2
(η)
imϕ
e
with
f
1,2
(ζ)=
-1/2
ζ
M
n
1,2
+
|m|+1
2
,
|m|
2
(ζ/n)
. Here
M
is a Whittaker function and
n=
n
1
+
n
2
+|m|+1
, equal to the principal quantum number. Contour plots for the real part of the wavefunctions in the
x,z
-plane are shown, including the values
ϕ=0
and
π
. The nucleus is represented as a black dot. The corresponding energy eigenvalues are given by
E
n
=-
1
2
2
n
, independent of other quantum numbers (in the field-free nonrelativistic case).
The hydrogen atom in a constant electric field ℰ along the
z
direction is also separable in parabolic coordinates and can thus be used to treat the Stark effect. The functions
f
1
(ξ)
and
f
2
(η)
are more complicated but can be obtained by perturbation expansions. To first order, the Stark effect energies are given by
E
n
n
1
n
2
≈-
1
2
2
n
+
3
2
ℰn(
n
1
-
n
2
)
​
. One atomic unit of electric field ℰ is equivalent to
5.142×
11
10
V/m. The presence of an electric field is shown by a red arrow.