130_notes.dvi

Thecoefficient of each power ofρmust be zero, so we can derive therecursion relationfor
the constantsak.

ak+1

ak

=

k+ℓ+ 1−λ

(k+ℓ+ 1)(k+ℓ) + 2(k+ℓ+ 1)−ℓ(ℓ+ 1)

=

k+ℓ+ 1−λ

k(k+ 2ℓ+ 1) + 2(k+ℓ+ 1)

=

k+ℓ+ 1−λ

k(k+ 2ℓ+ 2) + (k+ 2ℓ+ 2)

=

k+ℓ+ 1−λ

(k+ 1)(k+ 2ℓ+ 2)

→

1

k

This is then the power series for
G(ρ)→ρℓeρ

unless it somehow terminates. We canterminate the seriesif for some value ofk=nr,

λ=nr+ℓ+ 1≡n.

The number of nodes inGwill benr. We will callnthe principal quantum number, since the energy
will depend only onn.

Plugging in forλwe get theenergy eigenvalues.

Zα

√

−μc^2

2 E

=n.

E=−

1

2 n^2

Z^2 α^2 μc^2

Thesolutionsare

Rnℓ(ρ) =ρℓ

∑∞

k=0

akρke−ρ/^2.

The recursion relation is

ak+1=

k+ℓ+ 1−n

(k+ 1)(k+ 2ℓ+ 2)

ak.

We can rewriteρ, substituting the energy eigenvalue.

ρ=

√

− 8 μE

̄h^2

r=

√

4 μ^2 c^2 Z^2 α^2

̄h^2 n^2

r=

2 μcZα

̄hn

r=

2 Z

na 0

r

16.3.2 Computing the Radial Wavefunctions*

The radial wavefunctions are given by

R(ρ) =ρℓ

n−∑ℓ− 1

k=0

akρke−ρ/^2