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

 $eq2823.jpg$ (12.97)

for the Perelomov coherent state and

 $eq2824.jpg$ (12.98)

for the Barut–Girardello coherent state

 $eq2825.jpg$ (12.99)

for the atomic coherent state.

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

 $eq2826.jpg$ (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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