Spin and Addition of Angular Momentum Type Operators
8.3 Addition of angular momentum like quantities
Diagonalization of Hamiltonians like (8.20) or (8.21) requires us to combine two operators to a new operator according to j D l C s or j D j1C j1, respectively. From the perspective of spectroscopy, terms like (8.20) or (8.21) are the very reason why we have to know how to combine two angular momentum type operators in quantum mechanics. Diagonalization of (8.20) and (8.21) is important for understanding the spectra of atoms and molecules, and spin-orbit coupling also affects energy levels in materials. Furthermore, Hamiltonians of the form 2Js1 s2provide an effective description of interactions in magnetic materials, see Section17.7, and they are important for spin entanglement and spintronics. The advantage of introducing the combined angular momentum operator j D l C s is that it also satisfies angular momentum commutation rules (7.21)Œja; jb D i„ abcjcand therefore should have eigenstates jj; mji,
j2jj; mji D „2j.j C 1/jj; mji; jzjj; mji D „mjjj; mji: (8.22) However, j commutes with l2and s2,Œja; l2 D Œja; s2 D 0, and therefore we can try to construct the states in (8.22) such that they also satisfy the properties
l2jj; mj; `; si D „2`.` C 1/jj; mj; `; si;
s2jj; mj; `; si D „2s.s C 1/jj; mj; `; si:
The advantage of these states is that they are eigenstates of the coupling operator (8.20),
l sjj; mj; `; si D j2 l2 s2
2 jj; mj; `; si
D „2 j.j C 1/ `.` C 1/ s.s C 1/
2 jj; mj; `; si; (8.23) and therefore the energy shifts from spin-orbit coupling in these states are
E D 0e2„2
16m2hr3iŒj.j C 1/ `.` C 1/ s.s C 1/ : (8.24) The states that we know for the operators l and s are the eigenstates j`; m`i for l2 and lz, and js; msi for s2and sz, respectively. We can combine these states into states j`; m`i ˝ js; msi j`; m`I s; msi (8.25) which will be denoted as a tensor product basis of angular momentum states. The understanding in the tensor product notation is that l only acts on the first factor and s only on the second factor. Strictly speaking the combined angular momentum operator should be written as
jD l ˝ 1 C 1 ˝ s;
8.3 Addition of angular momentum like quantities 165
which automatically ensures the correct rule
j.j`; m`i ˝ js; msi/ D lj`; m`i ˝ js; msi C j`; m`i ˝ sjs; msi;
but we will continue with the standard physics notation j D l C s.
The main problem for combination of angular momenta is how to construct the eigenstates jj; mj; `; si for total angular momentum from the tensor products (8.25) of eigenstates of the initial angular momenta,
jj; mj; `; si D X
m`;ms
j`; m`I s; msih`; m`I s; msjj; mj; `; si: (8.26)
We will denote the states jj; mj; `; si as the combined angular momentum states.
There is no summation over indices`0 ¤ ` or s0 ¤ s on the right hand side because all states involved are eigenstates of l2 and s2 with the same eigenvalues
„2`.` C 1/ or „2s.s C 1/, respectively.
The components h`; m`I s; msjj; mj; `; si of the transformation matrix from the initial angular momenta states to the combined angular momentum states are known as Clebsch-Gordan coefficients or vector addition coefficients. The nota-tion h`; m`I s; msjj; mj; `; si is logically satisfactory by explicitly showing that the Clebsch-Gordan coefficients can also be thought of as the representation of the combined angular momentum states jj; mj; `; si in the basis of tensor product states j`; m`I s; msi. However, the notation is also redundant in terms of the quantum numbers ` and s, and a little clumsy. It is therefore convenient to abbreviate the notation by setting
h`; m`I s; msjj; mj; `; si h`; m`I s; msjj; mji:
The new angular momentum eigenstates must also be normalizable and orthog-onal for different eigenvalues, i.e. the transformation matrix must be unitary,
X
m`;ms
hj; mjj`; m`I s; msih`; m`I s; msjj0; m0ji D ıj;j0ımj;m0j; (8.27) X
j;mj
h`; m`I s; msjj; mjihj; mjj`; m0`I s; m0si D ım`;m0`ıms;m0s: (8.28)
The hermiticity properties
jzD.lzC sz/C; j˙D.lC s/C imply with the definition (4.31) of adjoint operators the relations
mjh`; m`I s; msjj; mji D.m`C ms/ h`; m`I s; msjj; mji (8.29)
and q
j.j C 1/ mj.mj˙ 1/h`; m`I s; msjj; mj˙ 1i
Dp
`.` C 1/ m`.m` 1/h`; m` 1I s; msjj; mji Cp
s.s C 1/ ms.ms 1/h`; m`I s; ms 1jj; mji: (8.30) Equation (8.29) yields
h`; m`I s; msjj; mji D ım`Cms;mjh`; m`I s; msjj; m`C msi:
The highest occurring value of mjwhich is also the highest occurring value for j is therefore`Cs, and there is only one such state. This determines the state j`Cs; `C s; `; si up to a phase factor to
j` C s; ` C s; `; si D j`; `I s; si; (8.31) i.e. we choose the phase factor as
h`; `I s; sj` C s; ` C si D 1:
Repeated application of j D lCson the state (8.31) then yields all the remaining states of the form j` C s; mj; `; si or equivalently the remaining Clebsch-Gordan coefficients of the form h`; m`I s; msj`Cs; mjD m`Cmsi with `s mj< `Cs.
For example, the next two lower states with j D` C s are given by jj` C s; ` C s; `; si Dp
2.` C s/j` C s; ` C s 1; `; si Dp
2`j`; ` 1I s; si Cp
2sj`; `I s; s 1i and
j2j` C s; ` C s; `; si D 2p
` C sp
2.` C s/ 1j` C s; ` C s 2; `; si D 2p
`.2` 1/j`; ` 2I s; si C 4p
`sj`; ` 1I s; s 1i C 2p
s.2s 1/j`; `I s; s 2i: (8.32) However, we have two states in the j`; m`I s; msi basis with total magnetic quantum number` C s 1, but so far discovered only one state in the jj; mj; `; si basis with this magnetic quantum number. We can therefore construct a second state with mjD
` C s 1, which is orthogonal to the state j` C s; ` C s 1; `; si,
j`Cs1; `Cs1; `; si D
r s
` C sj`; `1I s; si
s `
` C sj`; `I s; s1i: (8.33)
8.3 Addition of angular momentum like quantities 167
Application of j2would show that this state has j D` C s 1, which was already anticipated in the notation. Repeated application of the lowering operator jon this state would then yield all remaining states of the form j` C s 1; mj; `; si with 1 ` s mj< ` C s 1, e.g.
p` C s 1j` C s 1; ` C s 2; `; si D s
s2` 1
` C s j`; ` 2I s; si
r
`2s 1
` C sj`; `I s; s 2i Cps `
` C sj`; ` 1I s 1; si: (8.34) We have three states with mj D ` C s 2 in the direct product basis, viz. j`; ` 2I s; si, j`; `I s; s 2i and j`; ` 1I s 1; si, but so far we have only constructed two states in the combined angular momentum basis with mj D ` C s 2, viz. j` C s; ` C s 2; `; si and j` C s 1; ` C s 2; `; si.
We can therefore construct a third state in the combined angular momentum basis which is orthogonal to the other two states,
j` C s 2; ` C s 2; `; si / j`; ` 1I s 1; si
j` C s; ` C s 2; `; sih` C s; ` C s 2; `; sj`; ` 1I s 1; si
j` C s 1; ` C s 2; `; sih` C s 1; ` C s 2; `; sj`; ` 1I s 1; si:
Substitution of the states and Clebsch-Gordan coefficients from (8.32) and (8.34) and normalization yields
j` C s 2; ` C s 2; `; si D
s .2` 1/.2s 1/
.2` C 2s 1/.` C s 1/j`; ` 1I s 1; si C
p`.2` 1/j`; `I s; s 2i p
s.2s 1/j`; ` 2I s; si
p.2` C 2s 1/.` C s 1/ : (8.35)
Application of jthen yields the remaining states of the form j` C s 2; mj; `; si.
This process of repeated applications of j and forming new states with lower j through orthogonalization to the higher j states terminates when j reaches a minimal value j D j` sj, when all .2` C 1/.2s C 1/ states j`; m`I s; msi have been converted into the same number of states of the form jj; mj; `; si. In particular, we observe that there are2 min.`; s/ C 1 allowed values for j,
j2 fj` sj; j` sj C 1; : : : ; ` C s 1; ` C sg: (8.36) The procedure to reduce the state space in terms of total angular momentum eigenstates jj; mj; `; si through repeated applications of j and orthogonalizations is lengthy when the number of states.2` C 1/.2s C 1/ is large, and the reader
will certainly appreciate that Wigner [42] and Racah4 have derived expressions for general Clebsch-Gordan coefficients. Racah derived in particular the following expression (see also [9,34])
h`; m`I s; msjj; mji D ım`Cms;mj
X2
D1
./
pp .2j C 1/ .` C s j/Š .j C ` s/Š .j C s `/Š .j C ` C s C 1/Š Š .` m` /Š .s C ms /Š
p.` C m`/Š .` m`/Š .s C ms/Š .s ms/Š .j C mj/Š .j mj/Š .j s C m`C /Š .j ` msC /Š .` C s j /Š
! :
(8.37) The boundaries of the summation are determined by the requirements
maxŒ0; s m` j; ` C ms j minŒ` C s j; ` m`; s C ms:
Even if we decide to follow the standard convention of using real Clebsch-Gordan coefficients, there are still sign ambiguities for every particular value of j in j`sj j `Cs. This arises from the ambiguity of constructing the next orthogonal state when going from completed sets of states jj0; mj0; `; si, j < j0 ` C s to the next lower level j, because a sign ambiguity arises in the construction of the next orthogonal state jj; j; `; si. For example, Racah’s formula (8.37) would give us the state j` C s 2; ` C s 2; `; si constructed before in equation (8.35), but with an overall minus sign.
Tables of Clebsch-Gordan coefficients had been compiled in the olden days, but nowadays these coefficients are implemented in commercial mathematical software programs for numerical and symbolic calculation, and there are also free online applets for the calculation of Clebsch-Gordan coefficients.