Time evolution of linear and generalized Heisenberg algebra nonlinear Pöschl-Teller coherent states

We analyze the time evolution of two kinds of coherent states for a particle in a Pöschl-Teller potential. We find a pair of canonically conjugate operators and compare the behavior of their time evolution for both coherent states. The nonlinear ones are more localized. The trajectory in the phase space of the mean values of these two operators is a kind of generalization of the Rose algebraic curves. The new pair of canonically conjugate variables leads to a fourth-order Schrödinger equation which has the same energy spectrum as the Pöschl-Teller system.

Figure 1

$\xi /\lambda$ and $(\rho \lambda /\hslash )$ phase-space cycles. In all graphs, $\hslash =1,\lambda =1.5,\phi =0$. (a) $r=0.05$; (b) $r=0.5$; (c) $r=1$; (d) $r=4$. $\xi /\lambda$ is the horizontal axis and $\rho \lambda /\hslash$ is the vertical axis.

Figure 2

Time evolution of the uncertainty relation for $r=0.5, \phi =0$, and $b/\hslash =1$. Continuous (blue) line refers to nonlinear coherent state and dashed (red) line refers to linear coherent state.

Figure 3

Time evolution of the uncertainty relation for $r=1, \phi =0$, and $b/\hslash =1$. Continuous (blue) line refers to nonlinear coherent state and dashed (red) line refers to linear coherent state.

Figure 4

Time evolution of the uncertainty relation for $r=1.5, \phi =0$, and $b/\hslash =1$. Continuous (blue) line refers to nonlinear coherent state and dashed (red) line refers to linear coherent state.

Figure 5

Behavior of the nonlinear coherent state maximum of uncertainty for increasing values of $r$. The continuous line is a guide to the eye.