Kinetic energy equipartition: A tool to characterize quantum thermalization

According to both Bohmian and stochastic quantum mechanics, the standard quantum mechanical kinetic energy can be understood as consisting of two hidden-variable components. One component is associated with the current (or Bohmian) velocity, while the other is associated with the osmotic velocity (or quantum potential), and they are identified with the phase and the amplitude, respectively, of the wave function. These two components are experimentally accessible through the real and imaginary parts of the weak value of the momentum postselected in position. In this paper, a kinetic energy equipartition is presented as a signature of quantum thermalization in closed systems. This means that the expectation value of the standard kinetic energy is equally shared between the expectation values of the squares of these two hidden-variable components. Such components cannot be reached from expectation values linked to typical Hermitian operators. To illustrate these concepts, numerical results for the nonequilibrium dynamics of a few-particle harmonic trap under random disorder are presented. Furthermore, the advantages of using the center-of-mass frame of reference for dealing with systems containing many indistinguishable particles are also discussed.

Figure 1

Expectation values from the dynamics in scenario D1, with $(p_{01},p_{02})=(0,0)$. (a)–(d) A few initial cycles under no disorder with a smaller $(x_{01},x_{02})=(-2,2)$; (e)–(h) full dynamics under disorder with a larger $(x_{01},x_{02})=(-20,20)$. (a) and (e) show energies: orthodox kinetic energy $\langle K\rangle$, harmonic confining energy $\langle V_H\rangle$, Coulomb repulsion energy $\langle V_I\rangle$, and total energy $\langle H\rangle [\langle V_I\rangle$ is magnified by 100 in (e)]. (b) and (f) show velocities: Bohmian velocity $\langle v\rangle$, position $\langle x\rangle$, and osmotic velocity $\langle u\rangle$. (c) and (g) show a comparison of the kinetic energies: orthodox $\langle K\rangle$, Bohmian $\langle K_B\rangle$, and quantum potential $\langle Q_B\rangle$ energies. (d) and (h) show correlations: momentum $C_{{\widehat{p}}_1,{\widehat{p}}_2}$, Bohmian $C_{v_1,v_2}$ and osmotic $C_{u_1,u_2}$ velocities, and position $C_{{\widehat{x}}_1,{\widehat{x}}_2}$. All quantities in atomic units.

Figure 2

Expectation values from the dynamics in scenario D2, with $(x_{01},x_{02})=(-2,2)$. (a)–(d) A few initial cycles under no disorder with a smaller $(p_{01},p_{02})=(4,4)$; (e)–(h) full dynamics under disorder with a larger $(p_{01},p_{02})=(20,20)$. (a) and (e) show energies: orthodox kinetic energy $\langle K\rangle$, harmonic confining energy $\langle V_H\rangle$, Coulomb repulsion energy $\langle V_I\rangle$, and total energy $\langle H\rangle [\langle V_I\rangle$ is magnified by 100 in (e)]. (b) and (f) show velocities: Bohmian velocity $\langle v\rangle$, position $\langle x\rangle$, and osmotic velocity $\langle u\rangle$. (c) and (g) show a comparison of the kinetic energies: orthodox $\langle K\rangle$, Bohmian $\langle K_B\rangle$, and quantum potential $\langle Q_B\rangle$ energies. (d) and (h) show correlations: momentum $C_{{\widehat{p}}_1,{\widehat{p}}_2}$, Bohmian $C_{v_1,v_2}$ and osmotic $C_{u_1,u_2}$ velocities, and position $C_{{\widehat{x}}_1,{\widehat{x}}_2}$. All quantities in atomic units.

Figure 3

Expectation values from the dynamics in scenario D3, with mixed $x_0, p_0$ influence. (a)–(d) A few initial cycles under no disorder with small $(x_{01},x_{02})=(-2,2)$ and small $(p_{01},p_{02})=(2,2)$; (e)–(h) full dynamics under disorder with a large $(x_{01},x_{02})=(-20,20)$ and a large $(p_{01},p_{02})=(20,20)$. (a) and (e) show energies: orthodox kinetic energy $\langle K\rangle$, harmonic confining energy $\langle V_H\rangle$, Coulomb repulsion energy $\langle V_I\rangle$, and total energy $\langle H\rangle [\langle V_I\rangle$ is magnified by 100 in (e)]. (b) and (f) show velocities: Bohmian velocity $\langle v\rangle$, position $\langle x\rangle$, and osmotic velocity $\langle u\rangle$. (c) and (g) show a comparison of the kinetic energies: orthodox $\langle K\rangle$, Bohmian $\langle K_B\rangle$, and quantum potential $\langle Q_B\rangle$ energies. (d) and (h) show correlations: momentum $C_{{\widehat{p}}_1,{\widehat{p}}_2}$, Bohmian $C_{v_1,v_2}$ and osmotic $C_{u_1,u_2}$ velocities, and position $C_{{\widehat{x}}_1,{\widehat{x}}_2}$. All quantities in atomic units.

Figure 4

Center-of-mass and relative energy expectation values for scenarios D1 [(a), (d), and (g)], D2 [(b), (e), and (h)], and D3 [(c), (f), and (i)]. (a)–(c) Kinetic energies $\langle K_c\rangle$ and $\langle K_r\rangle$ and confining potential energies $\langle V_{H,c}\rangle$ and $\langle V_{H,r}\rangle$; the respective total energies $E_c=\langle K_c\rangle +\langle V_{H,c}\rangle$ and $E_r=\langle K_r\rangle +\langle V_{H,r}\rangle$ are shown as black lines. $\langle V_{I,r}\rangle$ is not included. (d)–(f) Bohmian kinetic $\langle K_{B,c}\rangle$ and quantum potential $\langle Q_{B,c}\rangle$ energies in comparison with orthodox $\langle K_c\rangle$ center-of-mass energies. (e)–(h) Bohmian kinetic $\langle K_{B,r}\rangle$ and quantum potential $\langle Q_{B,r}\rangle$ energies in comparison with orthodox $\langle K_r\rangle$ relative energies. The legend in (c) also applies to (a) and (b), the legend in (f) also applies to (d) and (e), and the legend in (i) also applies to (g) and (h). All quantities in atomic units.