Dm′m(j)(−γ,−β,−α)=Dmm′(j)*(α,β,γ),(1.238)
∑mDmm′(j)*(α,β,γ)Dmm″(j)(α,β,γ)=δm′m″,(1.239)
∑mDm′m(j)*(α,β,γ)Dm″m(j)(α,β,γ)=δm′m″.(1.240)
The reduced rotation matrices d(j)(β) are real. Thus, their matrix elements, dmm′(j)(β), from equation (1.238), obey:
dm′m(j)(−β)=dmm′(j)(β).(1.241)
From the general expression for the matrix elements of dmm′(j)(β), equation (1.167), it follows that
(−1)m′−md−m′,−m(j)(β)=dm′m(j)(β)=(−1)m′−mdmm′(j)(β),(1.242)
and hence
Dm′m(j)(α,β,γ)=(−1)m′−mD−m′,−m(j)*(α,β,γ).(1.243)
1.13 The rotation of 〈jm∣
The rotation of 〈jm∣ involves an important phase factor. From the rotation of ∣jm〉 by D(j)(α,β,γ):
D(α,β,γ)∣jm〉=∑m′Dm′m(j)(α,β,γ)∣jm′〉,(1.244)
∴〈jm∣D†(α,β,γ)=∑m′Dm′m(j)*(α,β,γ)〈jm′∣.(1.245)
Then, from the complex conjugate of equation (1.243):
〈jm∣D†(α,β,γ)=∑m′(−1)m′−mD−m′,−m(j)(α,β,γ)〈jm′∣,(1.246)
and replacing −m′↔m′, −m↔m, and noting that the sum is over m′=−j,−j+1,…,j−1,j and so is unaffected,
∴〈j,−m∣D†(α,β,γ)=∑m′(−1)−m′+mDm′m(j)(α,β,γ)〈j,−m′∣,(1.247)
∴(−1)−m〈j,−m∣D†(α,β,γ)=∑m′(−1)−m′Dm′m(j)(α,β,γ)〈j,−m′∣,(1.248)
i.e. (−1)−m〈j,−m∣ transforms like ∣jm〉. It is conventional to multiply both sides of equation (1.248) by (−1)j and then (−1)j−m〈j,−m∣ transforms like ∣jm〉, and the phase is real.
1.14 The rotation of the Ylm(θ,ϕ)
The transformations of the Ylm(θ,ϕ) under rotation follow from equation (1.221)
〈nˆ∣lm〉=Ylm(θ,ϕ)=Ylm(nˆ),(1.249)
and
D(R)∣nˆ〉=∣n′ˆ〉;(1.250)
whence from
D(R−1)∣lm〉=∑m′∣lm′〉〈lm′∣D(R−1)∣lm〉,(1.251)
i.e.
D(R−1)∣lm〉=∑m′∣lm′〉Dm′m(l)(R−1),(1.252)
then
〈nˆ∣D(R−1)∣lm〉=∑m′〈nˆ∣lm′〉Dm′m(l)(R−1).(1.253)
But
〈nˆ∣D(R−1)=〈nˆ′∣,(1.254)
∴Ylm(nˆ′)=∑m′Ylm′(nˆ)Dmm′(l)*(R),(1.255)
or
Ylm(θR,ϕR)=∑m′Ylm′(θ,ϕ)Dmm′(l)*(R).(1.256)
Similarly, from equation (1.248)
(−1)−mYl,−m(θR,ϕR)=∑m′(−1)−m′Yl,−m′(θ,ϕ)Dmm′(l)*(R).(1.257)
1.15 Exercises
1.1. Explore the commutator properties ofT1=00000−i0i0,T2=00i000−i00,T3=0−i0i00000,(1.258)in comparison with SO(3) and SU(2), (3,R) and (2,C).
1.2. Show thatd32(β)=c3−3c2s3cs2−s33c2sc3−2cs2s3−2c2s3cs23cs2−s3+2c2sc3−2cs2−3c2ss33cs23c2sc3,(1.259)where c≔cosβ2, s≔sinβ2.
1.3. Show that the results of equation (1.167) agree with equation (1.229) for ϕ=α=0 and j=l=1,2, and 3.
1.4. Show thatdm′m(j)(β)=(−1)m′−mdmm′(j)(β).(1.260)[Hint: in the binomial expansion of equation (1.160), which results in equation (1.162) and eventually equation (1.167), reverse the positions of cosβ2a+†, sinβ2a−† and −sinβ2a+†, cosβ2a−†, i.e. express the expansion so that it contains a+†cosβ2j+m−l, etc.]
1.5. Show that for R=(0,β,0) the Y1μ(θ,ϕ), μ=0,±1 obey equation (1.256). [Hint: express the Y1μ(θ,ϕ) in terms of x, y and z (cf. equations (1.188), (1.191), (1.193), (1.202) and (1.203)), obtain (x,y,z)R using Ry(β), and show that d(1)(β) (equation (1.65)) transforms the Y1μ(θ,ϕ) into the Y1μ(θR,ϕR).]
1.16 Spin-12 particles; neutron interferometry
The constituents of matter—electrons, protons, and neutrons—all have intrinsic spin of 12ℏ. ‘Intrinsic’ is the term coined to convey the fact that the dynamics of spin does not occur in physical space. ‘Spin space’ is not accessible to the physicist in the sense that the spin of a particle cannot be changed: it is intrinsic to the particle. In fact, it is not known what spin is. It is only known what spin does, namely ‘couple’ to other spins and angular momenta such that it behaves as a j=12 representation of SU(2).
In the absence of other particles and when its own angular momentum is zero, the quantum mechanics of a spin-12 particle is almost trivial. It can exist in two possible states: ‘spin up’ and ‘spin down’. These are directional components of the spin vector and are usually defined by
sˆzs=12,ms=±12=±12ℏs=12,ms=±12,(1.261)
where the direction is defined to be the z-axis in (3,R). However, there is one extraordinary property of spin-12 particles: a rotation through 2π does not leave their state kets unchanged! This is seen immediately from equation (1.52) for ϕ=2π, whence (using ∣sms〉↔χ±)
D12(nˆ,2π)χ±=Iˆcosπ−iσ⃗·nˆsinπχ±,(1.262)
∴D12(nˆ,2π)χ±=−χ±.(1.263)
This property is not observable where expectation values are involved; but it has a dramatic effect on the interferometry of beams of spin-12 particles.
The interferometry (diffractive splitting