# 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