Interacting classical and quantum particles

We apply Hall and Reginatto's theory of interacting classical and quantum ensembles to harmonically coupled particles, with a view to understanding its experimental implications. This hybrid theory has no free parameters and makes distinctive predictions that should allow it to be experimentally distinguished from quantum mechanics. It also bears on the questions of quantum measurement and quantum gravity.

Figure 1

Schematic illustration of the physical system considered in this paper. The spheres represent the particles, and the arrow the single dimension to which they are confined. The dashed line between the particles represents their harmonic interaction. Otherwise, they are free to move.

Figure 2

Error ellipses for the covariance matrix $Z$, shown at quarter-period intervals over two oscillation periods. The solid black ellipses are for the hybrid case and the dashed blue (gray) ellipses are for the quantum case. The axes are the dimensionless coordinates $\tilde{q}=q/x_0$ and $\tilde{x}=x/x_0$, where $x_0$ is the harmonic oscillator length $x_0={(\hslash /\mu {\omega }_{\mu })}^{1/2}$. First row: $t=0,T/4,T/2$. Second row: $t=3T/4,T,5T/4$. Third row: $t=3T/2,7T/4,2T$. Here $T=2\pi /{\omega }_{\mu }$ is the oscillator period. The initial conditions are given in the text, with $K_{xx}=K_{qq}=1/x_0^2$. In units of the reduced mass $\mu$, the masses are $m_q=m_x=2\mu$.

Figure 3

Dimensionless coordinate covariances in center-of-mass and relative coordinates versus dimensionless time $\tilde{t}=t{\omega }_{\mu }$. The hybrid oscillator curves are in black and the quantum oscillators curves in blue (gray). The solution is the same as in Fig. 2. Solid lines: $\mathrm{Var}\left(R\right)=Z_{RR}^{\prime }/x_0^2$. Dashed lines: $\mathrm{Var}\left(r\right)=Z_{rr}^{\prime }/x_0^2$. Dotted line: $\mathrm{Cov}(R,r)=Z_{Rr}^{\prime }/x_0^2$. In the quantum case, $\mathrm{Cov}(R,r)=0$.

Figure 4

Hybrid oscillator center-of-mass kinetic energy $E_R$ (solid line) and internal energy $E_I=E_r+\langle V\rangle$ (dashed line) versus dimensionless time $\tilde{t}=t{\omega }_{\mu }$. The graphed energies are made dimensionless by dividing by $\hslash {\omega }_0$. Masses are equal, $m_q=m_x$. The total energy is $E=1.125\hslash {\omega }_{\mu }$. The vertical dotted lines are at quarter periods. The contribution of the classical motion to the internal energy has not been included. The solution is the same as in Fig. 2. In the case of quantum or classical physics, $E_R$ and $E_I$ are constants.

Figure 5

Hybrid oscillator center-of-mass kinetic energy $E_R$ (solid line) and internal energy $E_I=E_r+\langle V\rangle$ (dashed line) versus dimensionless time $\tilde{t}=t{\omega }_{\mu }$. The graphed energies are made dimensionless by dividing by $\hslash {\omega }_0$. (a) Dominant classical mass $m_x=20m_q=21\mu$. (b) Dominant quantum mass $m_q=20m_x=21\mu$. The vertical dotted lines are at quarter periods. The contribution of the classical motion to the internal energy has not been included. Initial conditions and other parameters are the same as for Figs. 2, 3, and 4.

Figure 6

Hybrid oscillator center-of-mass kinetic energy $E_R$ (solid line), internal energy $E_I=E_r+\langle V\rangle$ (dashed line), and potential energy $\langle V\rangle$ (dotted line) versus dimensionless time $\tilde{t}=t{\omega }_{\mu }$. The graphed energies are made dimensionless by dividing by $\hslash {\omega }_0$. Masses are equal. The total energy is $E=12.51\hslash {\omega }_{\mu }$. The vertical dotted lines are at quarter periods. The contribution of the classical motion to the internal energy has not been included. Initial conditions and other parameters are the same as for Figs. 2, 3, and 4, except that $K_{xx}=K_{qq}=100/x_0^2$.