WOLFRAM|DEMONSTRATIONS PROJECT

Hydrogen-Like Continuum Eigenstates

​
Z
1
2
3
4
5
ℓ
0
1
2
3
4
k
1
show bound state
R
ℓ+1 ℓ
The positive-energy continuum states of a hydrogen-like system are described by the eigenfunctions
ψ
kℓm
​
(r,θ,ϕ)=
R
kℓ
(r)
Y
ℓm
(θ,ϕ)
with corresponding eigenvalues
E
k
=
2
k
2
, (
0≤k<∞
).
Y
ℓm
(θ,ϕ)
are the same spherical harmonics that occur for the bound states. In atomic units
ℏ=μ=e=1
, the radial equation can be written
-
1
2
′′
R
kℓ
(r)-
1
r
′
R
kℓ
(r)+
ℓ(ℓ+1)
2
2
r
-
Z
r

R
kℓ
(r)=
2
k
2
R
kℓ
(r)
. The solutions with the appropriate analytic and boundary conditions have the form
R
kℓ
(r)=
2k
πZ/2k
e
(2ℓ+1)!
Γ(ℓ+1+iZ/k)
ℓ
(2kr)
-ikr
e
1
F
1
(ℓ+1+iZ/k,2ℓ+2,2ikr)
. These functions are deltafunction-normalized, such that
∞
∫
0
R
kℓ
(r)
R
k'ℓ
(r)
2
r
dr=δ(k-k')
. They have the same functional forms (apart from normalization constants) as the discrete eigenfunctions under the substitution
n-iZ/k
.
You can plot the continuum function for various choices of
Z
, ℓ, and
k
. The asymptotic form for large
r
approaches a spherical wave of the form
R
kℓ
(r)≈
1
r
coskr+
Z
k
ln2kr-
π
2
(ℓ+1)-argΓ(ℓ+1+iZ/k)
. The terms in the argument of cos, in addition to
kr
, represent the phase shift with respect to the free particle. Coulomb scattering generally involves a significant number of partial waves.
A checkbox lets you compare the lowest-energy bound state with the same ℓ, that is, the wavefunctions for
1s
,
2p
,
3d
, ⋯ (magnified by a factor of
2ℓ+2
for better visualization). The
E=0
solution, obtained as the limit of the discrete function as
n∞
or of the continuum function as
k0
, has the form of a Bessel function:
R
0ℓ
(r)=const
J
2ℓ+2
(
8Zr
)
2Zr
.