Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 25

The functions

χn(z)≔znn!(1.270)

provide an orthonormal basis for expanding functions realised on z-space (the complex plane), with scalar products defined in terms of z-space integrals with Bargmann measure³, e−∣z∣2π [3]. This space is called Bargmann space.

The relevance of these functions to coherent states is implicit in the normalized coherent state form ∣z〉I, i.e. (cf. Volume 1, section 5.5)

∣z〉I≔e−∣z∣22∑n=0∞(z*)nn!∣n〉.(1.271)

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)).