Semiquantum versus semiclassical mechanics for simple nonlinear systems

Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behavior of expectation values of simple observables and of eigenvalues of the Groenewold operator are calculated numerically and compared for the various semiclassical and semiquantum approximations.

Figure 1

(Color online) Density plots showing the classical evolution of an initial Gaussian density centered at ${\alpha }_0=q_0=0.5$, with $\kappa =2$, as generated by the Hamiltonian $H=H_0^2∕E$. The parameters $m$, $\omega$, $E$ have been set equal to 1, and the times of the plots are, from left to right and top to bottom, $t=\pi ∕4$, $t=\pi ∕2$, $t=3\pi ∕4$, and $t=\pi$.

Figure 2

(Color online) Quantum evolution of an initial Gaussian Wigner function, with the same parameter values used in Fig. 1, and shown at the same times. Regions on which the pseudodensity becomes negative are shown in white.

Figure 3

(Color online) Classical and quantum evolution of the first and second moments of $\alpha =(q+ip)∕\sqrt{2}$ as generated by $H=H_0^2∕E$. Points on the classical curves are marked by + and points on the quantum curves by $\times$. Again $m=\omega =E=1$, and here $q_0=\sqrt{2}$, $p_0=0$. The first moments are graphed on the left and the second moments on the right. In the two upper graphs, $\kappa =2$, $\mu =1∕2$, ${\alpha }_0=1∕2$ and in the lower two graphs, $\kappa =1$, $\mu =1∕4$, ${\alpha }_0=1∕\sqrt{(}2)$.

Figure 4

(Color online) Comparison of first moments of $\alpha =(q+ip)∕\sqrt{2}$ for classical, semiquantum, quantum, and semiclassical evolutions generated by the Hamiltonian $H=H_0^3∕E^2$. Points on the classical, quantum, semiclassical, and semiquantum curves are labeled by $+$, $\times$, $엯$, and $◻$, respectively. The evolution is over the time interval $[0,\pi ]$ and again $m=\omega =E=1$, with $\kappa =2$, $\mu =1∕2$, ${\alpha }_0=0.5$.

Figure 5

(Color online) Identical to the preceding figure except that $\hslash$ has been reduced by a factor of 2, by setting $\kappa =1$, $\mu =1∕4$, ${\alpha }_0=1∕\sqrt{2}$.

Figure 6

(Color online) Comparison of largest two and least two eigenvalues for, from left to right and top to bottom, classical, semiquantum, and semiclassical evolutions generated by $H=H_0^3∕E^2$, for the time interval $[0,\pi ]$, and with $m=\omega =E=1$ and $\kappa =2$, $\mu =1∕2$, ${\alpha }_0=0.5$. The evolution of the first moment is reproduced from Fig. 5 in the graph at the bottom right for comparison. Each of the other graphs also features the quantum spectrum $\{0,1\}$ and in all graphs the values at the time points $t=1,2,3$ are marked $◇$ (classical), $엯$(semiclassical), $▵$ (quantum), and $◻$ (semiquantum).

Figure 7

(Color online) Comparison of square-negativity for classical, semi-quantum, and semiclassical evolutions generated by $H=H_0^3∕E^2$ for the time interval $[0,\pi ]$, and with $m=\omega =E=1$ and $\kappa =2$, $\mu =1∕2$, ${\alpha }_0=0.5$. The solid line, the solid line with $\times$ points, and the solid line with + points represent the classical, semiquantum, and semiclassical values, respectively.