the angles are: 0⩽α<2π, 0⩽β<π, 0⩽γ<2π. This results in an ambiguity for the rotation β=0, (α,0,γ)≡(α′,0,γ′) if α+γ=α′+γ′: this is referred to as ‘gimbal lock’ (where ‘gimbal’ refers to the rotation device or mechanical operator). This figure is adapted from that found on the Easyspin website.
Figure 1.1 depicts the following:
R(α,β,γ)=Rz(γ)RY(β)Rz¯(α).(1.1)
Note the order of the three rotations. The problem is that these three rotations are about axes belonging to three different frames of reference. The three rotations on the right-hand side of equation (1.1) can be restated in terms of a single frame of reference using similarity transformations, specifically
Rz(γ)=RY(β)Rz¯(γ)RY−1(β)(1.2)
and
RY(β)=Rz¯(α)Ry¯(β)Rz¯−1(α).(1.3)
Thus,
∴R(α,β,γ)=RY(β)Rz¯(γ)Rz¯(α);(1.5)
and, since Rz¯(γ) and Rz¯(α) commute,
∴R(α,β,γ)=RY(β)Rz¯(α)Rz¯(γ),(1.6)
∴R(α,β,γ)=Rz¯(α)Ry¯(β)Rz¯(γ).(1.8)
Note the new order of the three rotations (cf. equation (1.1)).
1.2 Matrix representations of spin and angular momentum operators
The matrix elements of Jˆz,Jˆ± in the {∣jm〉;j=0,12,1,32,…;m=+j,+j−1,…,−j} basis are (cf. Volume 1, chapter 11):
〈j′m′∣Jˆz∣jm〉=mℏδj′jδm′m,(1.9)
〈j′m′∣Jˆ±∣jm〉=(j∓m)(j±m+1)ℏδj′jδm′m±1.(1.10)
Matrix elements of Jˆx and Jˆy follow from:
Jˆx=12(Jˆ++Jˆ−),(1.11)
Jˆy=12i(Jˆ+−Jˆ−),(1.12)
where, recall Jˆ±≔Jˆx±iJˆy. Thus, the matrix representations of Jˆx,Jˆy, and Jˆz in a ∣jm〉 basis are:
Jˆx↔ℏ200000011000002020202000000300302002030030⋱,(1.13)
Jˆy↔ℏ2i0000001−10000020−2020−2000000300−30200−20300−30⋱,(1.14)
Jˆz↔ℏ200000100−100020000000−200003000010000−10000−3⋱.(1.15)
Note the ‘block-diagonal’ form of Jˆx and Jˆy. These blocks correspond to j=0,12,1,32,…. The matrix representation of Jˆz is diagonal with eigenvalues 0;12ℏ,−12ℏ;ℏ,0,−ℏ;32ℏ,12ℏ, −12ℏ,−32ℏ;…. It is normal practice to reduce these (infinite) matrices by breaking apart the blocks to give finite dimensional matrices. Thus, e.g.
j=12:Jˆx12↔ℏ20110,Jˆy12↔ℏ20−ii0,Jˆz12↔ℏ2100−1;(1.16)
j = 1:Jˆx(1)↔ℏ2020202020,Jˆy(1)↔ℏ20−2i02i0−2i02i0,Jˆz(1)↔ℏ220000000−2.(1.17)
The terminology:
Jˆx12≔Sˆx,Jˆy12≔Sˆy,Jˆz12≔Sˆz,(1.18)
σx≔0110,σy≔0−ii0,σz≔100−1,(1.19)
(cf. equation (1.16)), where σx,σy,σz are the Pauli spin matrices, is in common use.
1.3 The Pauli spin matrices
The Pauli spin matrices, σx,σy,σz, possess a number of useful properties. We redefine them by σj,σk,σl,(j,k,l)=(x,y,z). Then
σj2=σk2=σl2=Iˆ,(1.20)
σjσk+σkσj=0,forj≠k,(1.21)
i.e.
{σj,σk}=2δjkIˆ,(1.22)
where ‘{, }’ is an anticommutator bracket (also written ‘ [,]+’). Further,
[σj,σk]=2iεjklσl,(1.23)
where
εjkl≡εklj≡εljk≡1;εkjl≡εjlk≡εlkj≡−1.(1.24)
From equations (1.22) and (1.23)
σjσk=−σkσj=iσl.(1.25)
Also,
σj†=σj,(1.26)
det(σj)=−1,(1.27)
tr(σj)=0.(1.28)
For the three-dimensional Cartesian vector a⃗, σ⃗·a⃗ is a 2 × 2 matrix2:
σ⃗·a⃗≔σxax+σyay+σzaz,(1.29)
∴σ⃗·a⃗=azax−iayax+iay−az.(1.30)
This leads to the important identity:
(σ⃗·a⃗)(σ⃗·b⃗)=a⃗·b⃗Iˆ+iσ⃗·(a⃗×b⃗).(1.31)
This can be obtained from equations (1.22) and (1.23):
(σ⃗·a⃗)(σ⃗·b⃗)=∑jσjaj∑kσkbk.(1.32)
∴(σ⃗·a⃗)(σ⃗·b⃗)=∑j∑k12{σj,σk}+12[σj,σk]ajbk,(1.33)
∴(σ⃗·a⃗)(σ⃗·b⃗)=∑j∑k(δjk+iεjklσl)ajbk,(1.34)
∴(σ⃗·a⃗)(σ⃗·b⃗)=a⃗·b⃗Iˆ+iσ⃗·(a⃗×b⃗).(1.35)
If the components of a⃗ are real then
(σ⃗·a⃗)2=∣a⃗∣2Iˆ,(1.36)
where ∣a⃗∣ is the magnitude of the vector a⃗.
1.4 Matrix representations of rotations in ket space
We are
