where the Baker–Campbell–Hausdorff lemma is used (cf. Volume 1, chapter 5, equation (5.110)). Essentially all operators of relevance can be expressed in terms of a and a†, whence: for O=a
and from ∂∂z(eza)=aeza
⇒Γ(a)=∂∂z.(1.284)
For O=a†
Note:
1.∂∂z,z=1,cf.[a,a†]=1.(1.286)
2. z and ∂∂z are Hermitian adjoints for scalar products defined on Bargmann measure:e.g.forΨa=∑nanzn,Ψb=∑nbnzn,(1.287)∫∫dze−∣z∣2πΨa*∂∂zΨb=∑nan*bn+1(n+1)!=∫∫dze−∣z∣2πΨb(zΨa)*.(1.288)
1.18 Coherent states for SU(2)
The generalisation of the coherent state concept from the one-dimensional harmonic oscillator (Volume 1, section 5.5) to angular momentum is effected through their respective algebras: the Heisenberg–Weyl algebra in one dimension, hw(1) and su(2).
| hw(1) | su(2) | |
| Generators | a† | J+ |
| a | J− | |
| I | J 0 | |
| Commutator relations | [a,a†]=I | [J−,J+]=−2J0 |
| [I,a†]=0 | [J0,J+]=+J+ | |
| [I,a]=0 | [J0,J−]=−J− | |
| Lowest-weight state | ∣0〉 | ∣j,−j〉≔∣−j〉 |
| a∣0〉=0 | J−∣−j〉=0 |
Generalising the type-I coherent state from HW(1) to SU(2)
∣ζI〉≔expζ*J+−ζJ−∣−j〉,(1.289)
for ζ≔12θeiϕ,
eζ*J+−ζJ−=e−iθ(Jxsinϕ−Jycosϕ)=e−iθ(J⃗·nˆ),(1.290)
where nˆ is a unit vector in the x,y plane making an angle ϕ with the negative y-axis. This is illustrated in figure 1.5. All physically significant rotations are accommodated by this formalism (the apparent exclusion of rotations about the z-axis only excludes changes in phase, which could be introduced using e−iχJ0).
Figure 1.5. A depiction of the parameters ϕ and θ that define a type-I SU(2) coherent state.
The state ∣ζ〉I, ζ=ζ(θ,ϕ), can be expressed:
∣ζ〉I=∣θ,ϕ〉I=e−iθ(J⃗·nˆ)∣j,−j〉=∑m∣jm〉〈jm∣e−iθ(J⃗·nˆ)∣j,−j〉=∑m∣jm〉Dm,−j(j)(ϕ,θ,0)*.(1.291)
From the orthonormality of the D functions, sections 1.11 and 1.12,
I=(2j+1)4π∫dΩ∣θ,ϕ〉II〈θ,ϕ∣,dΩ=sinθdθdϕ.(1.292)
The states exp{ζ*J+−ζJ−}∣j,−j〉 are sometimes called ‘atomic coherent’ or ‘Bloch’ states (see, e.g. [4]).
The type-II coherent states of HW(1) can be generalised to SU(2):
∣z〉II≔exp(z*J+)∣j,−j〉.(1.293)
(∣ζ〉I and ∣z〉II are no longer trivially related, hence the use of z and ζ.)
The SU(2) states can be expressed in terms of the ∣z〉II:
∣Ψ〉→Ψ(z)=II〈z∣Ψ〉=〈−j∣ezJ−∣Ψ〉≔Ψj(z).(1.294)
Operators are mapped into z-space realisations, Γ(O) by
O∣Ψ〉→Γ(O)ΨJ(z)=〈z∣O∣Ψ〉=〈−j∣ezJ−O∣Ψ〉=〈−j∣(ezJ−Oe−zJ−)ezJ−∣Ψ〉=〈−j∣(O+[zJ−,O]+12[zJ−,[zJ−,O]]+⋯)ezJ−∣Ψ〉.(1.295)
Essentially all operators of relevance can be expressed in terms of J−, J0, and J+, whence: for O=J−
and from ∂∂z(ezJ−)=J−ezJ−
⇒Γ(J−)=∂∂z.(1.297)
For O=J0
and from z∂∂z(ezJ−)=zJ−ezJ−
⇒Γ(J0)=−j+z∂∂z.(1.299)
For O=J+
⇒Γ(J+)=2jz−z2∂∂z.(1.301)
Then
Γ(J0)znn!=(−j+n)znn!,n=0,1,2,…,(1.302)
Γ(J+)znn!=(2j−n)zn+1n!=(2j−n)n+1zn+1(n+1)!,(1.303)
Γ(J−)znn!=nzn−1n!=nzn−1(n−1)!.(1.304)
Comments
1 Starting from the state ∣j,−j〉, the raising action of Γ(J+) terminates at n=2j (as it should for SU(2)).
2 The representation is non-unitary, i.e. Γ(J+)†≠Γ(J−) for scalar products defined on Bargmann measure (for which (∂∂z)†=z).
3 This type of non-unitary representation is an example of a Dyson representation [5].
4 The ∣z〉II basis is defined on an SU(2) irrep labelled by j, i.e. it is a linear combination of the states ∣jm〉, m=j,j−1,…,−j.
1.19 Properties of SU(2) from coherent states
To obtain the properties of a quantum system which possesses an algebraic structure requires that unitary representations of the operators be found. This can be done in different ways. One way (which will not be developed here) is to change the measure of the z-space to enforce orthonormality. Thus, for the atomic coherent states:
I=∫∫dz(2j+1)π(1+∣z∣2)2j+2∣z〉II〈z∣(1.305)