Hydrogen-Like Continuum Eigenstates
Hydrogen-Like Continuum Eigenstates
The positive-energy continuum states of a hydrogen-like system are described by the eigenfunctions (r,θ,ϕ)=(r)(θ,ϕ) with corresponding eigenvalues =, (). (θ,ϕ) are the same spherical harmonics that occur for the bound states. In atomic units , the radial equation can be written . The solutions with the appropriate analytic and boundary conditions have the form (r)=Γ(ℓ+1+iZ/k)(ℓ+1+iZ/k,2ℓ+2,2ikr). These functions are deltafunction-normalized, such that (r)(r)dr=δ(k-k'). They have the same functional forms (apart from normalization constants) as the discrete eigenfunctions under the substitution .
ψ
kℓm
R
kℓ
Y
ℓm
E
k
2
k
2
0≤k<∞
Y
ℓm
ℏ=μ=e=1
-(r)-(r)+-(r)=(r)
1
2
′′
R
kℓ
1
r
′
R
kℓ
ℓ(ℓ+1)
2
2
r
Z
r
R
kℓ
2
k
2
R
kℓ
R
kℓ
2k
πZ/2k
e
(2ℓ+1)!
ℓ
(2kr)
-ikr
e
1
F
1
∞
∫
0
R
kℓ
R
k'ℓ
2
r
n-iZ/k
You can plot the continuum function for various choices of , ℓ, and . The asymptotic form for large approaches a spherical wave of the form (r)≈coskr+ln2kr-(ℓ+1)-argΓ(ℓ+1+iZ/k). The terms in the argument of cos, in addition to , represent the phase shift with respect to the free particle. Coulomb scattering generally involves a significant number of partial waves.
Z
k
r
R
kℓ
1
r
Z
k
π
2
kr
A checkbox lets you compare the lowest-energy bound state with the same ℓ, that is, the wavefunctions for , , , ⋯ (magnified by a factor of for better visualization). The solution, obtained as the limit of the discrete function as or of the continuum function as , has the form of a Bessel function: (r)=const(.
1s
2p
3d
2ℓ+2
E=0
n∞
k0
R
0ℓ
J
2ℓ+2
8Zr
)2Zr