130_notes.dvi

23.3 Examples

23.4 Derivations and Computations

23.4.1 The Relativistic Correction

Moving from the non-relativistic formula for the energy of an electron to the relativistic formula we
make the change

mc^2 +

p^2 e

2 m

→

(

p^2 c^2 +m^2 c^4

) 1 / 2

=mc^2

(

1 +

p^2 c^2

m^2 c^4

) 1 / 2

.

Taylor expanding the square root aroundp^2 = 0, we find

(

p^2 c^2 +m^2 c^4

) 1 / 2

=mc^2 +

1

2

p^2 c^2

mc^2

−

1

8

p^4 c^4

m^3 c^6

+···≈mc^2 +

p^2

2 m

−

p^4

8 m^3 c^2

So we have our next order correction term. Notice that p

2
2 mwas just the lowest order correction to
mc^2.

What about the “reduced mass problem”? The proton is very non-relativistic so only the electron
term is important and the reduced mass is very close to the electronmass. We can therefore neglect
the small correction to the small correction and use

H 1 =−

1

8

p^4 e

m^3 c^2

23.4.2 The Spin-Orbit Correction

We calculate the classical Hamiltonian for the spin-orbit interaction which we will later apply as a
perturbation. The B field from the proton in the electron’s rest frame is

B~=−~v

c

×E.~

Therefore the correction is
H 2 =
ge
2 mc

S~·B~=− ge

2 mc^2

S~·~v×E~

=

ge

2 m^2 c^2

S~·~p×∇~φ.

φonly depends onr⇒∇φ= ˆrdφdr=~rrdφdr

H 2 =

ge

2 m^2 c^2

S~·~p×~r^1

r

dφ

dr

=

−ge

2 m^2 c^2

~S·L~^1

r

dφ

dr

φ=

e

r

⇒

dφ

dr

=−

e

r^2

H 2 =

1

2

ge^2

2 m^2 c^2 r^3

~L·S~