Whence, consider
K≔∫∫dz∣z〉II〈z∣=∫∫dze−∣z∣2∑n(z*)nn!∣n〉∑mzmm!〈m∣;(1.272)
which, for z=reiϕ, gives
K=∫0∞rdr∫02πdϕe−r2∑n,mei(m−n)ϕrn+mn!m!∣n〉〈m∣.(1.273)
Now,
∫02πdϕei(m−n)ϕ=2πδmn,(1.274)
∴K=∑n∫0∞drr2n+1e−r22πn!∣n〉〈n∣=∑nΓ(n+1)22πn!∣n〉〈n∣=π∑n∣n〉〈n∣=πI.(1.275)
Thus, the resolution of the identity on Bargmann space is:
I=∫∫dzπ∣z〉II〈z∣≔∫∫dze−∣z∣2π∣z〉IIII〈z∣,(1.276)
where ∣z〉II↔χn(z), cf. equation (1.270). Then,
〈Ψ1∣Ψ2〉=∫∫dze−∣z∣2π〈Ψ1∣z〉IIII〈z∣Ψ2〉=∫∫dze−∣z∣2πΨ1*(z)Ψ2(z)=∫∫dμ(z)Ψ1*(z)Ψ2(z),(1.277)
where
Ψ(z)≔II〈z∣Ψ〉,(1.278)
dμ(z)≔e−∣z∣2πdz.(1.279)
Bargmann representations of functions are transformed into position representations of functions by the Bargmann transformation,
Ψ(x)=∫∫dμ(z)A(x,z*)Ψ(z),(1.280)
where
A(x,z*)≔1π14exp−12x2+2xz*−12(z*)2(1.281)
is the Bargmann kernel function.
Comments:
1 The orthogonality of the χn(z) is evident in a polar coordinate representation which gives (z*)nzm→ei(m−n)ϕ and ∫02πdϕei(m−n)ϕ=2πδmn.
2 The normalizability of the χn(z) is evident from the Gaussian form of Bargmann measure which ‘quenches’ the scalar products for large ∣z∣. (Indeed, the scalar products involve ‘camouflaged’ Hermite polynomials.)
3 The functions χn(z) are trivially generalised to tensor product functions,χn1(z1)⊗χn2(z2)⊗⋯which yields functions∑n1,n2,…αn1,n2,…z1n1n!!z2n2n2!⋯(cf. equations (1.147) and (1.176)).
1.17.1 Representation of operators
Consider the operator O and its representation, O↔Γ(O) in terms of z and ∂∂z, O(z,∂∂z) acting on z-space wave functions, Ψ(z). This is similar to the procedure presented in Volume 1, chapter 8, where, e.g. the operator px (momentum in the x direction) was shown to have a ‘position’ representation px↔−iℏ∂∂x when acting on Cartesian-space wave functions Ψ(x,y,z). The key there is to define a position eigenket basis {∣x〉} and arrive at statements such as 〈x∣pˆx∣Ψ〉=−iℏ∂∂x〈x∣Ψ〉=−iℏ∂Ψ(x)∂x. Thus, we proceed with the ∣z〉II basis, (∣z〉II≔ez*a†∣0〉, cf. Volume 1, section 5.5, equation (5.118))
O∣Ψ〉⇒ΓOz,∂∂zΨ(z)=II〈z∣O∣Ψ〉=〈0∣ezaO∣Ψ〉=〈0∣(ezaOe−za)eza∣Ψ〉=〈0∣O+[za,O]+12[za,[za,O]]+⋯eza∣Ψ〉,(1.282)
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).
