Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 27

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)