8.3 Addition of angular momentum like quantities
In classical mechanics, angular momentum is an additive vector quantity which is conserved in rotationally symmetric systems. Furthermore, the transformation equation (8.15) for spinor states involved addition of two different operators which both satisfy the angular momentum Lie algebra (7.21). However, before immersing ourselves into the technicalities of how angular momentum type operators are combined in quantum mechanics, it is worthwhile to point out that interactions in atoms and materials provide another direct physical motivation for addition of angular momentum like quantities.
We have seen in Section7.1that relative motion of two interacting particles with an interaction potentialV.x1 x2/can be described in terms of effective single particle motion of a (quasi)particle with locationr.t/ D x1.t/x2.t/, mass m D m1m2=.m1Cm2/, momentumpD.m2p1m1p2/=.m1Cm2/and angular momentum lDrp.
Furthermore, ifm2 m1, but the chargeq2is not much larger thanq1and the spinjs2jis not much larger thanjs1j, then we can assign a charge3qDq1and a spin sDs1to the quasiparticle with massm'm1.
A particle of chargeeand massmwith angular momentum operatorsland spin sexperiences a contribution to its energy levels from an interaction term
HlsD 0e2
8m2r3ls (8.20)
in its Hamiltonian, if it is moving in the electric fieldE D Ore=.40r2/of a much heavier particle of chargee. One can think of Hls as a magnetic dipole-dipole interaction. 0=4r3/ls, but finally it arises as a consequence of a relativistic generalization of the Schrửdinger equation. We will see this in Chapter 21, in particular equation (21.117). However, for the moment we simply accept the existence of terms like (8.20) as an experimental fact. These terms contribute to the fine structure of spectral lines. The term (8.20) is known as aspin-orbit coupling term orlscoupling term, and applies in this particular form to the energy levels of the quasiparticle which describes relative motion in a two-particle system. However, if there are many charged particles like in a many-electron atom, then there will also be interaction terms between angular momenta and spins of different particles in the system, i.e. we will have terms of the form
Hj1j2 Df.r12/j1 j2; (8.21) wherejiare angular momentum like operators. We will superficially denote all these operators (including spin) simply as angular momentum operators in the following.
3We will return to the question of assignment of charge and spin to the quasiparticle for relative motion in Section18.4.
Diagonalization of Hamiltonians like (8.20) or (8.21) requires us to combine two operators to a new operator according tojDlCsorjDj1Cj1, 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 form2Js1s2provide 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 lCs is that it also satisfies angular momentum commutation rules (7.21)Œja;jb D i„abcjcand therefore should have eigenstatesjj;mji,
j2jj;mji D „2j.jC1/jj;mji; jzjj;mji D „mjjj;mji: (8.22) However,jcommutes withl2ands2,Œja;l2DŒja;s2D0, and therefore we can try to construct the states in (8.22) such that they also satisfy the properties
l2jj;mj; `;si D „2`.`C1/jj;mj; `;si; s2jj;mj; `;si D „2s.sC1/jj;mj; `;si:
The advantage of these states is that they are eigenstates of the coupling operator (8.20),
lsjj;mj; `;si D j2l2s2
2 jj;mj; `;si
D „2 j.jC1/`.`C1/s.sC1/
2 jj;mj; `;si; (8.23) and therefore the energy shifts from spin-orbit coupling in these states are
ED 0e2„2
16m2hr3iŒj.jC1/`.`C1/s.sC1/ : (8.24) The states that we know for the operatorslandsare the eigenstatesj`;m`iforl2 andlz, andjs;msifors2andsz, respectively. We can combine these states into states j`;m`i ˝ js;msi j`;m`Is;msi (8.25) which will be denoted as a tensor product basis of angular momentum states. The understanding in the tensor product notation is thatl only acts on the first factor andsonly on the second factor. Strictly speaking the combined angular momentum operator should be written as
jDl˝1C1˝s;
8.3 Addition of angular momentum like quantities 165
which automatically ensures the correct rule
j.j`;m`i ˝ js;msi/Dlj`;m`i ˝ js;msi C j`;m`i ˝sjs;msi; but we will continue with the standard physics notationjDlCs.
The main problem for combination of angular momenta is how to construct the eigenstatesjj;mj; `;sifor 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`Is;msih`;m`Is;msjj;mj; `;si: (8.26)
We will denote the statesjj;mj; `;sias the combined angular momentum states.
There is no summation over indices`0 ¤ `or s0 ¤ son the right hand side because all states involved are eigenstates ofl2 ands2 with the same eigenvalues
„2`.`C1/or„2s.sC1/, respectively.
The components h`;m`Is;msjj;mj; `;si of the transformation matrix from the initial angular momenta states to the combined angular momentum states are known as Clebsch-Gordan coefficientsor vector addition coefficients. The nota- tionh`;m`Is;msjj;mj; `;siis logically satisfactory by explicitly showing that the Clebsch-Gordan coefficients can also be thought of as the representation of the combined angular momentum statesjj;mj; `;siin the basis of tensor product states j`;m`Is;msi. However, the notation is also redundant in terms of the quantum numbers `ands, and a little clumsy. It is therefore convenient to abbreviate the notation by setting
h`;m`Is;msjj;mj; `;si h`;m`Is;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`Is;msih`;m`Is;msjj0;m0ji Dıj;j0ımj;m0j; (8.27) X
j;mj
h`;m`Is;msjj;mjihj;mjj`;m0`Is;m0si Dım`;m0`ıms;m0s: (8.28)
The hermiticity properties
jzD.lzCsz/C; j˙D.lCs/C imply with the definition (4.31) of adjoint operators the relations
mjh`;m`Is;msjj;mji D.m`Cms/h`;m`Is;msjj;mji (8.29)
and q
j.jC1/mj.mj˙1/h`;m`Is;msjj;mj˙1i Dp
`.`C1/m`.m`1/h`;m`1Is;msjj;mji Cp
s.sC1/ms.ms1/h`;m`Is;ms1jj;mji: (8.30) Equation (8.29) yields
h`;m`Is;msjj;mji Dım`Cms;mjh`;m`Is;msjj;m`Cmsi:
The highest occurring value ofmjwhich is also the highest occurring value forjis therefore`Cs, and there is only one such state. This determines the statej`Cs; `C s; `;siup to a phase factor to
j`Cs; `Cs; `;si D j`; `Is;si; (8.31) i.e. we choose the phase factor as
h`; `Is;sj`Cs; `Csi D1:
Repeated application ofj DlCson the state (8.31) then yields all the remaining states of the formj`Cs;mj; `;sior equivalently the remaining Clebsch-Gordan coefficients of the formh`;m`Is;msj`Cs;mjDm`Cmsiwith`smj< `Cs.
For example, the next two lower states withjD`Csare given by jj`Cs; `Cs; `;si Dp
2.`Cs/j`Cs; `Cs1; `;si Dp
2`j`; `1Is;si Cp
2sj`; `Is;s1i and
j2j`Cs; `Cs; `;si D2p
`Csp
2.`Cs/1j`Cs; `Cs2; `;si D2p
`.2`1/j`; `2Is;si C4p
`sj`; `1Is;s1i C2p
s.2s1/j`; `Is;s2i: (8.32) However, we have two states in thej`;m`Is;msibasis with total magnetic quantum number`Cs1, but so far discovered only one state in thejj;mj; `;sibasis with this magnetic quantum number. We can therefore construct a second state withmjD
`Cs1, which is orthogonal to the statej`Cs; `Cs1; `;si, j`Cs1; `Cs1; `;si D
r s
`Csj`; `1Is;si s `
`Csj`; `Is;s1i: (8.33)
8.3 Addition of angular momentum like quantities 167
Application ofj2would show that this state hasj D`Cs1, which was already anticipated in the notation. Repeated application of the lowering operatorjon this state would then yield all remaining states of the form j`Cs1;mj; `;si with 1`smj< `Cs1, e.g.
p`Cs1j`Cs1; `Cs2; `;si D s
s2`1
`Cs j`; `2Is;si
r
`2s1
`Csj`; `Is;s2i Cps`
`Csj`; `1Is1;si: (8.34) We have three states with mj D ` C s 2 in the direct product basis, viz. j`; ` 2Is;si, j`; `Is;s 2i and j`; ` 1Is 1;si, but so far we have only constructed two states in the combined angular momentum basis with mj D `C s2, viz. j`Cs; `Cs2; `;si and j`C s1; ` Cs2; `;si.
We can therefore construct a third state in the combined angular momentum basis which is orthogonal to the other two states,
j`Cs2; `Cs2; `;si / j`; `1Is1;si
j`Cs; `Cs2; `;sih`Cs; `Cs2; `;sj`; `1Is1;si
j`Cs1; `Cs2; `;sih`Cs1; `Cs2; `;sj`; `1Is1;si: Substitution of the states and Clebsch-Gordan coefficients from (8.32) and (8.34) and normalization yields
j`Cs2; `Cs2; `;si D
s .2`1/.2s1/
.2`C2s1/.`Cs1/j`; `1Is1;si C
p`.2`1/j`; `Is;s2i p
s.2s1/j`; `2Is;si
p.2`C2s1/.`Cs1/ : (8.35)
Application ofjthen yields the remaining states of the formj`Cs2;mj; `;si.
This process of repeated applications of j and forming new states with lowerj through orthogonalization to the higherjstates terminates whenjreaches a minimal valuejD j`sj, when all.2`C1/.2sC1/statesj`;m`Is;msihave been converted into the same number of states of the formjj;mj; `;si. In particular, we observe that there are2min.`;s/C1allowed values forj,
j2 fj`sj;j`sj C1; : : : ; `Cs1; `Csg: (8.36) The procedure to reduce the state space in terms of total angular momentum eigenstatesjj;mj; `;sithrough repeated applications ofj and orthogonalizations is lengthy when the number of states.2`C1/.2sC 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`Is;msjj;mji Dım`Cms;mj
X2
D1
./
pp .2jC1/.`Csj/Š.jC`s/Š.jCs`/Š .jC`CsC1/ŠŠ.`m`/Š.sCms/Š
p.`Cm`/Š.`m`/Š.sCms/Š.sms/Š.jCmj/Š.jmj/Š .jsCm`C/Š.j`msC/Š.`Csj/Š
! :
(8.37) The boundaries of the summation are determined by the requirements
maxŒ0;sm`j; `Cmsj minŒ`Csj; `m`;sCms: Even if we decide to follow the standard convention of using real Clebsch- Gordan coefficients, there are still sign ambiguities for every particular value ofjin j`sj j`Cs. This arises from the ambiguity of constructing the next orthogonal state when going from completed sets of statesjj0;mj0; `;si,j < j0 `Csto the next lower levelj, because a sign ambiguity arises in the construction of the next orthogonal statejj;j; `;si. For example, Racah’s formula (8.37) would give us the statej`Cs2; `Cs2; `;siconstructed 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.