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Part B: Many-Particle Angular Momentum Operators.

The commutation relations determine the properties of the angular momentum and spin

operators. They are completely analogous:

, , .

L L i L etc

L L iL

L L L L L

L L L L

x y z

x y

z z

z z

=

= ±

= + −

= + +

±

+ −

− +

2 2

2

, , .

S S i S etc

S S iS

S S S S S

S S S S

x y z

x y

z z

z z

=

= ±

= + −

= + +

±

+ −

− +

2 2

2

The one-electron eigenfunctions for the , L Lz2 operators are the spherical harmonics

2 2

1 1 1

1 1 1

ˆ ( , ) ( 1) ( , )ˆ ( , ) ( , )ˆ ( , ) ~ ( , ) ( 1) ( 1) ( , ) ( , ) ( , )ˆ ( , ) ~ ( , ) ( 1) ( 1) ( , ) ( , ) ( , )

m ml l

m mz l l

m m m ml l l l

m m m ml l l l

L Y l l Y

L Y m Y

L Y Y l l m m Y C l m Y

L Y Y l l m m Y C l m Y

θ ϕ θ ϕ

θ ϕ θ ϕ

θ ϕ θ ϕ θ ϕ θ ϕ

θ ϕ θ ϕ θ ϕ θ ϕ

+ + ++ +

− − −− −

= +

=

= + − + ≡

= + − − ≡

h

h

h

h

Here I included the precise proportionality constants. For later convenience they are

abbreviated as ( , )C l m± . The one-electron spin eigenfunctions are denoted as

Y Y1 21 2

1 21 2

//

//;= =−α β. Explicitly the various equations read

( )

, ,

S

S S Sz

2 2 212121 3

412

0

α α α

α α α α β

= + =

= = =+ −

( )

, ,

S

S S Sz

2 2 212121 3

412

0

β β β

β β β α β

= + =

= − = =+ −

A single electron (a so-called spin 1/2 particle) is always described by the spin-functions

α and β . Higher than spin 1/2 functions show up in many-electron wave functions.

Nuclear spin operators (indicated I ) also satisfy precisely the same commutation

relations ˆ ˆ ˆ,x y zI I i I⎡ ⎤ =⎣ ⎦ h and cyclic permutations. Nuclei on the other hand can have

higher spins (they consist of protons and neutrons that individually are spin 1/2 particles).

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This means that the nuclear spin functions might for example be a triplet (I=1), or a

quartet (I=3/2). The mathematics underlying all of these different physical phenomena is

precisely the same, and we will focus on the case of angular momentum for an atom

consisting of a certain number of electrons.

The many-electron operators are defined in an analogous fashion

( ) ( ) ..., .L L L etcxtotal

x x= + +1 2 ( ) ( ) ..., .S S S etcxtotal

x x= + +1 2

The operators ,S S S S S S Stotalxtotal

xtotal

ytotal

ytotal

ztotal

ztotal2 = + + , (and similarly ) are

complicated (two-electron) operators that contain mixed terms like ( ) ( )S Sx x1 2 . We can

always express things in term of products of S Stotal total+ − etc, so we do not need to use S 2

directly. We can work with the sum-operators on a product of functions. As examples

consider ( ( ) ( )) [ ( ) ( )]( ( ) ( )) [( ( ) ( )) ( ) ( ) ( ) ( )]

( ) ( ) ( ) ( ) ( ) ( )

S S S S Sztotal

z z z zα α α α α α α α

α α α α α α

1 2 1 2 1 2 1 1 2 1 2 2

21 2

21 2 1 2

= + = +

= + =

and ( ( ) ( )) [ ( ) ( )]( ( ) ( )) [( ( ) ( )) ( ) ( ) ( ) ( )]

( ) ( ) ( ) ( )S S S S Stotal− − − − −= + = +

= +α α α α α α α α

β α α β1 2 1 2 1 2 1 1 2 1 2 2

1 2 1 2

hence we can think of acting with the one-electron operator on each one-electron function

separately and summing the result.

The procedure works in the same way if we use the Ltotal− operator on a product of

spherical harmonics. Let us take the p functions and abbreviate p Y p Y p Y1 11

0 10

1 11= = =−

−; ;

. Then

( ( ) ( )) ( ( ) ( ) ( ( ) ( ))L p p p p p ptotal+ − −= +0 1 1 1 0 01 2 2 1 2 1 2

( ( ) ( )) ( ) ( ) ( ) ( ) ( )L p p p p p pztotal

0 1 0 1 0 11 2 0 1 1 2 1 2− − −= − = −

For p-functions (l =1) the factor l l m m l m( ) ( ) ( , , )+ − + = = = −1 1 2 1 0 1 .

We also wish to examine what happens if we act on an antisymmetric Slater determinant

of spin-orbitals. Also here we can just act with the one-electron operator on each of the

,L total2

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product functions in the determinant and sum the result. The reason is that the sum

operator is symmetric and commutes with any permutation of electron labels, e.g.

[ ( ) ( )][ ( , ) ( ) ( )] [ ( ) ( )] ( ) ( )

( , )[( ( ) ( )) ( ) ( )] [ ( ) ( )] ( ) ( )

S S P S S

P S S S Sz z z z

z z z z

1 2 12 1 2 1 2 2 1

12 1 2 1 2 2 1 2 1

+ = +

= + = +

α β α β

α β α β

This argument is clearly general and so ˆ ˆ.... ( ....)total totalz a b z a bS Sϕ ϕ ϕ ϕ=

Let us consider the spin orbitals in a p-manifold: p p p p p p1 0 1 1 0 1, , , , ,− − . In this notation

p p p p1 1 1 1= =α β; , etc. Below I will give a set of examples of operations of spin and

angular momentum operations. You can verify the results, and see that the basic rules are

not very difficult.

,

( !), L p p L p p p p

S p p p p antisymmetry S p pz

z

+

−

= =

= = =1 1 1 1 1 1

1 1 1 1 1 1

0 2

0 0

The above relations suffice to show that p p1 1 is the ml = 2 component of the 1D

multiplet. You would need to use the form 2 2ˆ ˆ ˆ ˆ ˆz zL L L L L− += + +h . All other functions in the

multiplet can be generated by acting successively by L− . For example

( ) ( ) ( )

L p p p p p p

L p p p p p p p pz

− = +

+ = +1 1 0 1 1 0

0 1 1 0 0 1 1 0

2

Importantly, this eigenstate with eigenvalue lm = h is not a determinant but a linear

combination of determinants. In this state you cannot say that these orbitals are occupied

and the rest empty. The wave function is more complicated. You can act with the 2L

operator on this new function to show that the eigenvalue has not changed: 2(2+1) 2h is

the answer you should find. But it is more work to show this explicitly (give it a try). You

can keep applying the L− operator until you find the state 1 1p p− − ( 2lm = − h ), and acting

with L− once more yields zero.

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As another example ,

,

( ) ( )

L p p p p L p p p p

S p p S p p p p

L p p p p p p p p

S p p p p p p

z

z

+

+

− − −

−

= = =

= =

= + =

= +

1 0 1 1 1 0 1 0

1 0 1 0 1 0

1 0 0 0 1 1 1 1

1 0 1 0 1 0

2 0

0

2 2

The first two equations establish that p p1 0 is the m ml s= =1 1, component of the 3P

multiplet. Please verify. All other 9 states can be obtained by successive application of L−

and S− as illustrated by the last two examples given.

We will do some exercises with the angular momentum operators elaborating on some

examples as illustrated above. There is a vast literature on the topic and the above is a

very brief summary. Let me re-emphasize that angular momentum theory underlies NMR

spectra (nuclear spin functions). Electron spin resonance can be treated analogously. This

will be illustrated in class. To treat these phenomena we act on product of nuclear spin

functions. The principles are similar, and spin functions are in fact a bit easier because

the proportionality constants for spin 12

particles are unity rather than 2 . Let me also

mention that we can use a similar treatment to find eigenfunctions of the total angular

momentum operator ˆˆ ˆJ L S= + , which is particularly relevant for atomic term symbols.

We will go through some examples in class and in the problem set.

III. General decomposition of a product basis of angular momentum eigenfunctions

into eigenfunctions of the total angular momentum operators.

In this section we consider the construction of eigenfunctions of the angular momentum

operators for a composite particle (in fact it applies to any product function, for example

spin-orbitals 1pα ). The structure is very general. We will consider the products for two

particles, but from this you can construct products for an arbitrary number of particles

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using the same general principles. You have seen examples in NMR before we get to this

part of the lecture notes.

Consider the individual multiplets 1 1 1 1 1, , ...l m m l l= − and 2 2 2 2 2, , ...l m m l l= − which are

eigenfunctions of the angular momentum operators 2 ˆ(1), (1)zL L and 2 ˆ(2), (2)zL L

respectively. Here we are using the Dirac notation for states: 1 1,l m indicates an

eigenfunction characterized by the quantum numbers in the so-called ket. The full set of

product functions 1 1 2 2, ,l m l m spans a space of dimension 1 2(2 1)(2 1)l l+ + . We want to

decompose this product basis into a set of new basis functions that are eigenfunctions of

the composite angular momentum operators 2,

ˆ ˆ,total z totalL L , where ˆ ˆ ˆ(1) (2)totalL L L= + . Let us

indicate these eigenfunctions as , , ...L M M L L= − . Such a set of functions with a given

value of L is called a multiplet, and the dimension of the multiplet is 2L+1. We will see

below that the possible eigenvalues L range from 1 2 1 2 1 2, 1,...,L l l l l l l= + + − − , or,

assuming that 1 2l l≥ , we can also say that the multiplets that are occurring are

1 2 2, 0,..,2L l l k k l= + − = .

The total dimension of the space spanned by these multiplets is

2 2

1 2 2 1 20,2 0,2

2 1 2 2 2 2

2 1

{2( ) 1} (2 1)(2( ) 1) 2

1(2 1)(2 1) 2 (2 1) 2 (2 (2 1))2

(2 1)(2 1)

k l k ll l k l l l k

l l l l l l

l l

= =

+ − + = + + + −

= + + + + − +

= + +

∑ ∑

This is consistent with the total number of product functions, as should be the case.

Let us rationalize this result further by an explicit construction of the eigenfunctions.

Each product function itself is an eigenfunction of ,ˆz totL :

, 1 1 2 2 1 2 1 1 2 2ˆ , , ( ) , ,z totL l m l m m m l m l m= +h . Hence we can easily arrange the product

functions in a table, such that along a row, they all have the same value of M.

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Table: Arrangement of product functions according to M-values (eigenvalue of ,ˆz totL )

M= 1 2l l+ 1 1 2 2, ,l l l l …1

M= 1 2 1l l+ − 1 1 2 2, 1 ,l l l l− 1 1 2 2, , 1l l l l − … 2

M= 1 2 2l l+ − 1 1 2 2, 2 ,l l l l− 1 1 2 2, 1 , 1l l l l− − 1 1 2 2, , 2l l l l − … 3

………… ……………… ……………….. …………………

M= 1 2 2l l− − + 1 1 2 2, 2 ,l l l l− + − 1 1 2 2, 1 , 1l l l l− + − + 1 1 2 2, , 2l l l l− − + …3

M= 1 2 1l l− − + 1 1 2 2, 1 ,l l l l− + − 1 1 2 2, , 1l l l l− − + …2

M= 1 2l l− − 1 1 2 2, ,l l l l− − …1

# functions 1 22( ) 1l l+ + 1 22( 1) 1l l+ − + 1 22( 2) 1l l+ − +

The overall multiplet structure can be discerned now. In the second column, top row, we

find the function with the maximum value 1 2M L l l= = + . This function is also an

eigenfunction of 2ˆtotalL . This follows because , 1 1 2 2

ˆ , ,totalL l l l l+ =0. You can complete the

argument: why is it an eigenfunction of 2ˆtotalL ? The other functions in this multiplet are

generated by acting with ,ˆ ˆ ˆ(1) (2)totalL L L− − −= + , hence

, 1 1 2 2 1 1 1 1 2 2 2 2 1 1 2 2ˆ , , ( ( , ) , 1 , ( , ) , , 1 )totalL l l l l C l l l l l l C l l l l l l− − −= − + −

and so forth. It is seen that in general we find a linear combination of products states,

except for the highest and lowest possible M-value in the space. This process continues

as you ladder down and decrease the value of M , until 1 2M L l l= − = − − . The number of

functions is precisely the number of functions in the first column, 1 22( ) 1l l+ + . The next

multiplet is constructed by starting from the eigenfunction that has 1 2 1M l l= + − , and is

orthogonal to the linear combination function already found. Alternatively we might try

to find that linear combination such that acting with ,ˆtotalL+ on this combination yields 0.

Then it will be an eigenfunction of both 2ˆtotalL and ,

ˆz totalL . For example

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, 2 2 1 1 2 2 1 1 1 1 2 2

2 2 1 1 1 1 2 2 1 1 2 2

ˆ ( ( , 1) , 1 , ( , 1) , , 1 )

( , 1) ( , 1)[ , , , , ] 0totalL C l l l l l l C l l l l l l

C l l C l l l l l l l l l l+ + +

+ +

− − − − −

= − − − =.

This means it will be an eigenfunction of 2ˆtotalL with 1 2 1L l l= + − .

Even though we require a linear combination of the product states to construct the true

eigenfunctions, the number of functions in this second multiplet is precisely the same as

in the second column of the table, namely 1 22( 1) 1l l+ − + . The scheme now repeats itself.

To start constructing the next multiplet find the remaining highest M-state that is

orthogonal to the states already found and ladder down. Alternatively, and this is usually

easier, find that linear combination of functions of specific M-values such that acting

with ,ˆtotalL+ yields precisely 0.

The above scheme is completely general, and applies to a variety of problems in physics.

For example we can construct eigenfunctions of the spin-operators in this fashion, and in

this way we find the proper combinations of singlet, doublet, triplet spin functions. The

same strategy applies to nuclear spin functions. These problems are very instructive:

They give exact results because the basis set is complete, while the algebra is often not so

tedious. We can also couple different angular momentum operators for example to get

eigenfunctions of the total angular momentum ˆˆ ˆJ L S= + . These problems are always

completely analogous if you work on constructing many-particle functions. Essentially

this is true because the angular momentum operators for different particles commute. It is

also possible to use angular momentum theory to construct for example integrals and

orbitals of higher l-value. As an example you can evaluate integrals like ˆi jp x d . The

selection rules are easily evaluated (meaning you can know when an integral is zero). The

precise evaluation of non-zero integrals requires more work.

To help you digest this material there are some exercises!