130_notes.dvi

Enjmjℓs=En(0)+

En(0)

2

2 mc^2

[

3 −n

4 ∓j+^21

2

ℓ+^12

]

We can write (ℓ+^12 ) as (j+^12 ∓^12 ), so that

4 ∓j+^21

2

ℓ+^12

=

4 j+ 2∓ 2

(j+^12 )(j+^12 ∓^12 )

= 4

(j+^12 ∓^12 )

(j+^12 ∓^12 )(j+^12 )

and we get a nice cancellation giving us a simple formula.

Enlm=En(0)+E

(0)n^2

2 mc^2

[

3 −j^4 +n 1

2

]

This is independent ofℓso the states of different total angular momentum split in energy but there
is still a good deal of degeneracy.

add B-field

n=2

S1/2

P3/2 P1/2

S P1/2

1/2

P3/2

add relativity

add Darwin

• spin-orbit

We have calculated the fine structure effects in Hydrogen. There are, of course, other, smaller
corrections to the energies. A correction from field theory, the Lamb Shift, causes states of different
ℓto shift apart slightly. Nevertheless, the states of definite totalangular momentum are the energy
eigenstates until we somehow break spherical symmetry.