130_notes.dvi

1.20 Solutions to the Radial Equation for Constant Potentials

Solutions to the radial equation (See section 15.1) in a constant potential are important since they
are the solutions for largerin potentials of limitted range. They are therefore used in scattering
problems as the incoming and outgoing states. The solutions are thespherical Bessel and spherical
Neumann functions (See section 15.3).

jℓ(ρ) = (−ρ)ℓ

(

1

ρ

d

dρ

)ℓ

sinρ

ρ

→

sin(ρ−ℓπ 2 )

ρ

nℓ(ρ) =−(−ρ)ℓ

(

1

ρ

d

dρ

)ℓ

cosρ

ρ

→

−cos(ρ−ℓπ 2 )

ρ

whereρ=kr. The linear combination of these which falls off properly at largeris called theHankel
functionof the first type.

h(1)ℓ (ρ) =jℓ(ρ) +inℓ(ρ) = (−ρ)ℓ

(

1

ρ

d

dρ

)ℓ

sinρ−icosρ

ρ

→−

i

ρ

ei(ρ−

ℓπ 2 )

We use these solutions to do apartial wave analysis of scattering, solve forbound states of a
spherical potentialwell, solve for bound states of aninfinite spherical well(a spherical “box”),
and solve for scattering from a spherical potential well.

1.21 Hydrogen

The Hydrogen (Coulomb potential) radial equation (See section 16)is solved by finding the behavior
at larger, then finding the behavior at smallr, then using a power series solution to get

R(ρ) =ρℓ

∑∞

k=0

akρke−ρ/^2

withρ=

√

− 8 μE
̄h^2 r. To keep the wavefunction normalizable the power series must terminate, giving
us ourenergy eigenvaluecondition.

En=−

Z^2 α^2 mc^2

2 n^2

Herenis called theprinciple quantum numberand it is given by

n=nr+ℓ+ 1

wherenris the number of nodes in the radial wavefunction. It is an odd feature of Hydrogen that
a radial excitation and an angular excitation have the same energy.

So a Hydrogenenergy eigenstateψnℓm(~x) =Rnℓ(r)Yℓm(θ,φ) is described by three integer quantum
numbers with the requirements thatn≥1,ℓ < nand also an integer, and−l≤m≤ℓ. The ground
state of Hydrogen isψ 100 and has energy of -13.6 eV. We compute several of the lowest energy
eigenstates.

The diagram below shows the lowest energy bound states of Hydrogen and their typical decays.