Bohmian quantum trajectories from coherent states

We find that real and complex Bohmian quantum trajectories resulting from well-localized Klauder coherent states in the quasi-Poissonian regime possess qualitatively the same type of trajectories as those obtained from a purely classical analysis of the corresponding Hamilton-Jacobi equation. In the complex cases we treated the quantum potential results to a constant, such that the agreement is exact. For the real cases we provide conjectures for analytical solutions for the trajectories as well as the corresponding quantum potentials. The overall qualitative behavior is governed by the Mandel parameter determining the regime in which the wave functions evolve as solitonlike structures. We demonstrate these features explicitly for the harmonic oscillator and the Pöschl-Teller potential.

Figure 1

(a) Real Bohmian quantum trajectories as functions of time from Klauder coherent states (scattered) versus classical trajectories corresponding to (16) (solid lines). (b) Quantum potential from Klauder coherent states (scattered) versus conjectured formula (16) (solid lines). We have taken $x_0=2$ and computed the maxima to $x_{\text{max}}^{0.5}=1$, $x_{\text{max}}^1=1.4142$, $x_{\text{max}}^2=2$, $x_{\text{max}}^3=2.4495$. (In a.u.)

Figure 2

Complex Bohmian quantum trajectories as functions of time for different initial values $x_0$ resulting from stationary states ${\psi }_1(x,t)$ and ${\psi }_5(x,t)$ in (a) and (b), respectively. (In a.u.)

Figure 3

Complex Bohmian trajectories resulting from Klauder coherent states (scattered) compared to the purely classical computation (26) (solid) for different values of $J$, with initial value (a) $x_0=3+i$ and (b) $x_0=3-i$ with maximal values $x_{\text{max}}^{0.5}=1$, $x_{\text{max}}^1=1.4142$, $x_{\text{max}}^2=2$, $x_{\text{max}}^3=2.4495$. (In a.u.)

Figure 4

Real Bohmian trajectories as functions of time from Klauder coherent states (scattered) versus classical trajectories (solid lines) corresponding to (33). (a) Quasi-Poissonian distribution with initial value $x_0=2$, coupling constants $\kappa =90$, $\lambda =100$, maxima $x_{\text{m}}^2=2.0504447$, $x_{\text{m}}^{0.5}=2.0251224$, $x_{\text{m}}^{0.1}=2.0112100$, and periods $T^2=0.2612875$, $T^{0.5}=0.2622200$, $T^{0.1}=0.2627495$ for different values of $J$. (b) Quasi-Poissonian distribution with initial value $x_0=2$, coupling constants $\kappa =2$, $\lambda =3$, maxima $x_{\text{m}}^{0.0022906}=2.059522$, $x_{\text{m}}^{0.00057265}=2.0295876$, $x_{\text{m}}^{0.000114531}=2.0131884$, and periods $T^{0.0022906}=8.34795$, $T^{0.00057265}=8.36305$, $T^{0.000114531}=8.37129$ for different values of $J$. (c) Sub-Poissonian distribution with initial value $x_0=2$, coupling constants $\kappa =2$, $\lambda =3$ for $J=2,10,20$ (scattered) and $\kappa =9$, $\lambda =10$ for $J=20.2846$ (solid). (d) Sub-Poissonian distribution for various initial values with coupling constants $\kappa =2$, $\lambda =3$ for $J=2$. (In a.u.)

Figure 5

(a) Periodic solitonlike motion in the quasi-Poissonian regime for $\kappa =90$, $\lambda =100$ (thin) and $\kappa =2$, $\lambda =3$ (broad) with $Q=-0.000054529$ identical in both cases. (b) Spreading wave in the sub-Poissonian regime with $Q=-0.0917752$ for $J=5$ and $\kappa =2$, $\lambda =3$. (In a.u.)

Figure 6

Product of the position and momentum uncertainty as functions of time for different values of the Mandel parameter $Q$. (a) The coupling constants are $\kappa =2,\lambda =3$, with $J=0.0022906$ (red dotted), $J=0.00057265$ (black dashed), $J=0.000114531$ (blue solid), and $J=0.1$ for the subpanel with $\kappa =90,\lambda =100$. (b) The coupling constants are $\kappa =2,\lambda =3$ with $J=2$ (red dotted), $J=10$ (black dashed), and $J=50$ (blue solid). (In a.u.)

Figure 7

Complex Bohmian trajectories as functions of time for different initial values $x_0$ resulting from stationary states ${\Psi }_0(x,t)$ and ${\Psi }_5(x,t)$ in (a) and (b), respectively. (In a.u.)

Figure 8

Complex Bohmian trajectories as functions of time from Klauder coherent states (scattered) versus classical trajectories corresponding to solutions of (8) and (9) for the complex Pöschl-Teller Hamiltonian ${\mathcal{H}}_{\text{PT}}$ (38) (solid lines) for quasi-Poissonian distribution with $\kappa =90$, $\lambda =100$. (a) The evolution is shown from $t=0$ to $t=0.27$ with initial values on the isochrone with $t_f=0.04$ for $J=0.5$ and in (b) from $t=0$ to $t=0.5$ (quantum) and $t=0$ to $t=3.0$ (classical) with initial value $x_0=3-1.7i$ and $p_0=32.907+3.16416i$ for $J=20$. The energy for this trajectory is therefore $E={\mathcal{H}}_{\text{PT}}(x_0,p_0)=-8.44045+102.134i$. (In a.u.)

Figure 9

Complex Bohmian quantum trajectories as functions of time from $J=0.5$ Klauder coherent states (scattered) versus classical trajectories (solid lines) corresponding to solutions of (8) and (9) for the complex Pöschl-Teller potential $V_{\text{PT}}$ (a) real part, (b) imaginary part for quasi-Poissonian and localized distribution from $t=0$ to $t=1.78$ with $\kappa =90$, $\lambda =100$. The initial values are $x_0=3+1.5i$, $p_0=-30.1922+0.385121i$ such that $E=-6.55991-13.5182i$ (black solid, red scattered) and $x_0=4.5$, $p_0=41.8376i$ with real energy $E=-31.7564$ (blue solid, red scattered). (In a.u.)

Figure 10

Complex Bohmian trajectories as functions of time from $J=0.00057265$-Klauder coherent states (scattered) versus classical (solid lines) corresponding to solutions of (8) and (9) for the complex Pöschl-Teller potential $V_{\text{PT}}$ (a) real part, (b) imaginary part for the quasi-Poissonian regime and spread distribution from $t=0$ to $t=100$ (quantum) and $t=0$ to $t=32$ (classical) with $\kappa =2$, $\lambda =3$. The initial values are $x_0=3+1.5i$ and $p_0=-0.788329+0.157336i$ such that $E=-0.187539-0.0275087i$ (black solid, red scattered) and $x_0=2$, $p_0=-0.49446i$ with real energy $E=-0.277833$ (blue solid, red scattered). (In a.u.)

Figure 11

Complex Bohmian trajectories as functions of time from $J=10$ Klauder coherent states (scattered) versus classical (solid lines) corresponding to solutions of (8) and (9) for the complex Pöschl-Teller potential $V_{\text{PT}}$ (a) real part, (b) imaginary part for the sub-Poissonian regime from $t=0$ to $t=21$ (quantum) and $t=0$ to $t=32$ (classical) with $\kappa =2$, $\lambda =3$. The initial values are $x_0=2+0.2i$ and $p_0=-0.665052-0.406733i$ such that $E=-0.0941366-0.364635i$ (black solid, red scattered). (In a.u.)