130_notes.dvi

23.4.6 The Anomalous Zeeman Effect

We compute the energy change due to a weak magnetic field using first order Perturbation Theory.

〈

ψnℓjmj

∣

∣

∣

∣

eB

2 mc

(Lz+ 2Sz)

∣

∣

∣

∣ψnℓjmj

〉

(Lz+ 2Sz) =Jz+Sz

TheJzpart is easy since we are in eigenstates of that operator.
〈
ψnℓjmj

∣

∣

∣

∣

eB

2 mc

Jz

∣

∣

∣

∣ψnℓjmj

〉

=

eB

2 mc

̄hmj

TheSzis harder since we are not in eigenstates of that one. We need

〈

ψnℓjmj

∣

∣eB

2 mcSz

∣

∣ψnℓjmj〉, but

we don’t know howSzacts on these. So, we must write

∣

∣ψnjmjℓs〉in terms of|ψnℓmℓsms〉.

En(1) =

〈

ψnjℓmj

∣

∣

∣

∣

eB

2 mc

(Jz+Sz)

∣

∣

∣

∣ψnjℓmj

〉

=

eB

2 mc

(

mj ̄h+

〈

ψnjℓmj|Sz|ψnjℓmj

〉)

We already know how to write in terms of these states of definitemℓandms.

ψn(ℓ+ (^12) )ℓ(m+ (^12) ) = αYℓmχ++βYℓ(m+1)χ−
ψn(ℓ− (^12) )ℓ(m+ (^12) ) = βYℓmχ+−αYℓ(m+1)χ−
α =

√

ℓ+m+ 1

2 ℓ+ 1

β =

√

ℓ−m

2 ℓ+ 1

Let’s do thej=ℓ+^12 state first.

〈

ψnjℓmj|Sz|ψnjℓmj

〉

=

〈

αYℓ(mj− (^12) )χ++βYℓ(mj+ (^12) )χ−|Sz|αYℓ(mj− (^12) )χ++βYℓ(mj+ (^12) )χ−

〉

=

1

2

̄h

(

α^2 −β^2

)

m=mj−^12

Forj=ℓ−^12 ,
〈
ψnjℓmj|Sz|ψnjℓmj

〉

=

1

2

̄h

(

β^2 −α^2

)

m=mj−^12

We can combine the two formulas forj=ℓ±^12.

〈

ψnjℓmj|Sz|ψnjℓmj

〉

= ±

̄h

2

(

α^2 −β^2

)

=±

̄h

2

ℓ+m+ 1−ℓ+m

2 ℓ+ 1

= ±

̄h

2

2(mj−^12 ) + 1

2 ℓ+ 1

=±

mj ̄h

2 ℓ+ 1