Quantum-to-classical limit in a Hamiltonian system

The dynamics of quantum and classical probability distributions are computed for a model of two coupled rotors (or pendulums), with emphasis on the $\hslash$ scaling of the quantum-classical (QC) differences. This scaling is not the same for coarse-grained and fine-grained quantities. The QC differences in the averages of observables scale as ${\hslash }^{2∕3}$ for chaotic states, but as ${\hslash }^2$ for regular states. The QC differences in probability distributions scale as ${\hslash }^{1∕3}$ for chaotic states. No simple scaling is found for those differences in regular states, although their overall magnitudes are similar to those in chaotic states. QC differences arise first in the short-wavelength regime, and subsequently spread to all wavelengths. A satisfactory classical limit is obtained without invoking environmental decoherence, which in any case would be effective in suppressing only short-wavelength QC differences.

Figure 1

Quantum-classical difference for the average momentum, for two values of $\hslash$.

Figure 2

Maximum QC differences in average momentum $⟨p⟩$, rms momentum spread $\Delta p$, and momentum correlation $c_{12}=⟨p_1{p}_2⟩-⟨p_1⟩⟨p_2⟩$ for a chaotic state (left scale) and a regular state (right scale).

Figure 3

Angular momentum distributions for the quantum state (QM) and the classical ensemble (CE) in the chaotic regime, at $t=10$. Graphs on the left are for rotor 1; graphs on the right are for rotor 2. Top graphs are for $\hslash =0.02$; bottom graphs are for $\hslash =0.005$.

Figure 4

Integrated absolute differences between quantum and classical momentum distributions, $D$ [Eq. (5)]. Chaotic state on the left scale; regular state on the right scale.

Figure 5

Quantum and classical angular momentum distributions for a regular state at $t=50$. (a) $\hslash =0.01$. (b) $\hslash =0.005$. Only rotor 1 is illustrated, since the two rotors have very similar momentum distributions in this state.

Figure 6

Contour map of $\mid C_{M_1{M}_2}^Q-C_{M_1{M}_2}^C\mid$, the QC differences in the Fourier coefficients of the configuration-space distribution, for the chaotic state at $t=7.5$, $\hslash =0.01$. The contour levels range from 0.01 to 0.06.

Figure 7

Contour map of $\mid C_{M_1{M}_2}^Q-C_{M_1{M}_2}^C\mid$ for the chaotic state at $t=30$, $\hslash =0.01$. The contour levels range from 0.01 to 0.03.

Figure 8

rms average of the real and imaginary parts of $\mid C_{M_1{M}_2}^Q-C_{M_1{M}_2}^C\mid$ for $\mid M_1\mid +\mid M_2\mid ⩽20$, for the chaotic state.

Figure 9

rms average of the real and imaginary parts of $\mid C_{M_1{M}_2}^Q-C_{M_1{M}_2}^C\mid$ for $\mid M_1\mid +\mid M_2\mid ⩽20$, for the regular state.