Coherent States of the Generalized Heisenberg Algebra with $\varvec{k-1}$ Isolated States

Abstract

In this paper we construct the generalized Heisenberg algebra with $k-1$ isolated states. We use this algebra to construct three types of coherent states. Finally we discuss the non-classical properties related to the coherent states.

Appendix A

In this Appendix, we prove the Eq. (27) explicitly. The LHS of the Eq. (27) can be written as

$$\begin{aligned} LHS= & {} \sum _{n=0}^{\infty } \frac{ 1}{ n! \prod _{j=2}^k \{ j \}_n } |n\rangle \langle n | \int _0^{\infty } c_{k-1}^2 \mu (x) x^n dx =I \nonumber \\= & {} \sum _{n=0}^{\infty } \frac{ 1}{ n! [\prod _{j=2}^k \{ j \}_n ] {[}\prod _{j=2}^k (j-1)! ] } |n\rangle \langle n | \int _0^{\infty } G^{ k+1, 1}_{2, k+2} ( 0, k ; 0, 1, 2, \ldots , k, 0 | x ) x^n dx , \end{aligned}$$

(50)

If we use $\{ j \}_n ( j-1)! = \Gamma ( j + n )$ and

$$\begin{aligned} \int _0^{\infty } G^{ k+1, 1}_{2, k+2} ( 0, k ; 0, 1, 2, \ldots , k, 0 | x ) x^n dx = \frac{ [ \prod _{j=1}^{k+1} \Gamma ( b_j + n +1 ) ] \Gamma ( - a_1 - n ) }{ \Gamma ( - b_{k+2} -n ) \Gamma ( a_2 + n +1)} \end{aligned}$$

(51)

with

$$\begin{aligned} a_1 =0, \quad a_2 =k, \quad b_1 = 0, \quad b_2 = 1, \quad b_3 =2,\ldots , b_{k+1} = k, \quad b_{ k+2} =0, \end{aligned}$$

we have

$$\begin{aligned} \int _0^{\infty } G^{ k+1, 1}_{2, k+2} ( 0, k ; 0, 1, 2, \ldots , k, 0 | x ) x^n dx = n! \prod _{j=2}^{k} \Gamma ( n+j ) \end{aligned}$$

(52)

Inserting the Eq. (52) into the Eq. (50) , we have

$$\begin{aligned} LHS = \sum _{n=0}^{\infty } |n\rangle \langle n | = I \end{aligned}$$

(53)