Coherent states, displaced number states and Laguerre polynomial states for su(1,1) Lie algebra

The ladder operator formalism of a general quantum state for su(1, 1) Lie a lgebra is obtained. The state bears the generally deformed oscillator algeb raic structure. It is found that the Perelomov's coherent state is a su(1, 1) nonlinear coherent state. The expansion and the exponential form of the nonlinear coherent state are given. We obtain the matrix elements of the su (1, 1) displacement operator in terms of the hypergeometric functions and t he expansions of the displaced number states and Laguerre polynomial states are followed. Finally some interesting su(1, 1) optical systems are discus sed.