where we set ∂j = ∂/∂xj and xj = x+, xz, x. For example, we obtain after some calculation

μ;k|F(K+,Kz,K)|μ;k=F(A)(,z,)(1|μ|2)exp(xz/2)1(μ+x+)(μ+x)exp(xz)2kx=0,

(12.97)

for the Perelomov coherent state and

z;k|F(K+,Kz,K)|z;k=F(N)(z,z,z)exp(xz/2)I2k1(2|z|exp(xz/2))I2k1(2|z|)x=0,

(12.98)

for the Barut–Girardello coherent state

v;j|F(K+,Kz,K)|v;j=F(A)(,z,+)1+(μ+x+)(μ+x)exp(xz)(1+|μ|2)exp(xz/2)x=0,

(12.99)

for the atomic coherent state.

Next, we consider average values in a quantum state described by the density operator ρ

ρ=Z1exp(a+K++azKz+aK),

(12.100)

where Z = Tr exp(a+K+ + azKz + aK) with Tr being the trace operation. The parameters a± and az are required such that the density operator ρ

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