130_notes.dvi

Since our eigenstates ofJ^2 andJzare also eigenstates ofL^2 andS^2 (but notLzorSz), these
are ideal for computing the spin orbit interaction. In fact, they are going to be the true energy
eigenstates, as rotational symmetry tells us they must.

21.3 Adding Spin^12 to Integer Orbital Angular Momentum

Our goal is to add orbital angular momentum with quantum numberℓto spin^12. We can show in
several ways that, forℓ 6 = 0, that the total angular momentum quantum number hastwo possible
valuesj=ℓ+^12 orj=ℓ−^12. Forℓ= 0, onlyj=^12 is allowed. First lets argue that this makes
sense when we are adding twovectors. For example if we add a vector of length 3 to a vector of
length 0.5, the resulting vector could take on a length between 2.5 and 3.5 For quantized angular
momentum, we will only have the half integers allowed, rather than a continuous range. Also we
know that the quantum numbers likeℓare not exactly the length of the vector but are close. So
these two values make sense physically.

We can also count states for each eigenvalue ofJzas in the following examples.

• See Example 21.7.1:Counting states forℓ= 3 plus spin^12 .*


• See Example 21.7.2:Counting states for anyℓplus spin^12 .*
  As in the last section, we could start with the highestJzstate,Yℓℓχ+, and apply thelowering
  operatorto find the rest of the multiplet withj=ℓ+^12. This works well for some specificℓbut is
  hard to generalize.

We can work the problem in general. We know that each eigenstate ofJ^2 andJzwill be alinear
combination of the two product stateswith the rightm.

ψj(m+^12 )=αYℓmχ++βYℓ(m+1)χ−

audio

The coefficientsαandβmust bedetermined(see section 21.8.4)by operating withJ^2.

ψ(ℓ+ (^12) )(m+ (^12) )=

√

ℓ+m+ 1

2 ℓ+ 1

Yℓmχ++

√

ℓ−m

2 ℓ+ 1

Yℓ(m+1)χ−

ψ(ℓ− (^12) )(m+ (^12) )=

√

ℓ−m

2 ℓ+ 1

Yℓmχ+−

√

ℓ+m+ 1

2 ℓ+ 1

Yℓ(m+1)χ−

We have made a choice in how to write these equations: mmust be the same throughout. The
negativemstates are symmetric with the positive ones. These equations will beapplied when we
calculate thefine structure of Hydrogenand when we study theanomalous Zeeman effect.

21.4 Spectroscopic Notation

A common way to name states in atomic physics is to usespectroscopic notation. It is essentially
a standard way to write down the angular momementum quantum numbers of a state. Thegeneral
formisN^2 s+1Lj, whereNis the principal quantum number and will often be omitted,sis the
total spin quantum number ((2s+ 1) is the number of spin states),Lrefers to the orbital angular