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arXiv:2002.12035 · Molecular Physics

Mean square displacement of a free quantum particle in a thermal state

Roberto R. Marquardt

Published Sep 10, 2021• 9 citations
arXivHTMLPDF

Abstract

The mean square displacement x(t)−x(0)2 of the position x of a free particle of mass m at thermal equilibrium is evaluated quantum mechanically. An analytical expression is obtained which shows an initial quadratic increase of the mean square displacement with time and later on a linear growth, with the slope ℏ/m, quite at variance with the result from classical statistical mechanics. Results are discussed in relation to observables from helium scattering or spin-echo experiments, and their possible interpretation in terms of the classical and quantum mechanical expression for the mean square displacement of an essentially free particle.

DOI
10.1080/00268976.2021.1971315

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The mean square displacement x(t)−x(0)2 of the position x of a free particle of mass m at thermal equilibrium is evaluated quantum mechanically. An analytical expression is obtained which shows an initial quadratic increase of the mean square displacement with time and later on a linear growth, with the slope ℏ/m, quite at variance with the result from classical statistical mechanics. Results are discussed in relation to observables from helium scattering or spin-echo experiments, and their possible interpretation in terms of the classical and quantum mechanical expression for the mean square displacement of an essentially free particle.
A commonly used quantum mechanical formulation of the mean square displacement δx2(t) is based on the quantum correlation function of the position operator. While this quantity yields the classically expected result δx2(t)∝t2 for a ballistic particle in the interval topology, it is found here to diverge at essentially all times, when evaluated on infinitely large rings. A somewhat different formulation of the mean square displacement in quantum mechanics was proposed in a previous work (R. Marquardt, Mol. Phys. 119, e1971315 (2021)) and yielded the result δx2(t)∝t for the ballistic particle on an infinitely large ring. Here, it is shown analytically that that formulation yields δx2(t)≡0 for the ballistic particle on an infinitely long interval. The two formulations define two different, topology dependent quantities that have the dimension of an area and that could in principle be determined from the same experiment, following a measurement idea proposed in the aforementioned work. That idea is critically reviewed here.
The quantum-mechanical motion of a free particle coupled to a linearly dissipative environment is analyzed in the Hamiltonian formalism. The total Hamiltonian is diagonalized and its eigenstates displayed. These results are used to discuss the role of initial conditions on the subsequent motion. A widely used initial condition---heat bath uncoupled from particle---is compared with another one in which the initial off-diagonal coherence of the reduced density matrix is comparable to that in the state of thermal equilibrium of the coupled system. Different transient behaviors, on time scales longer than the inverse cutoff frequency of the bath are found. The mean-square momentum of the particle in the steady state is found to depend on this highest bath frequency. An analysis of the correlated states of particle and heat reservoir shows that the latter behaves in some sense like a position-measuring apparatus.
We propose a scheme for quantum brownian motion of a particle in a fermionic bath. Based on the spin coherent-state representation of the noise operators and a canonical thermal distribution of the associated c numbers, we derive a quantum analog of generalized Langevin equation for quantum-mechanical mean position of the particle subjected to an external force field. The approach allows us to map the quantum problem on a classical setting. The quantum dispersion around the mean can be estimated order by order by a set of quantum correction equations up to a desired degree of accuracy for a given nonlinear potential. We derive a quantum diffusion equation for free particle and show that quantization, in general, enhances the mean-square displacement. Increase in temperature leads to suppression of mean-square displacement. The method is based on canonical quantization procedure and may be used for understanding diffusive transport and thermally activated processes in a fermionic bath.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
In this note, we try to develop bound state quantum mechanics directly from statistical arguments. The key idea, we argue, is that a statistical treatment of a system considers “free particles” together with a stochastic equilibrium driving mechanism. If an acceleration driving potential exists (as in classical statistical mechanics), it alters the results, but does not change the main scenario of free particles interacting through a stochastic mechanism at a point because one does not follow particles in time, noting how they accelerate as they travel. (For example, a Maxwell-Boltzmann gas distribution is based on two body elastic scattering whether there is a potential or not. The potential affects the density, but not the relative momentum probabilities at any x.) For the quantum case, the potential V(x) is the stochastic equilibrium driver, but on average kinetic energy plus potential must equal total energy. Again, one does not follow the quantum particle in time trying to see if or how it accelerates.
2d quantum-mechanical problem of the time evolution of a particle in a quadratic potential is studied. We suppose that the center of the potential is displaced in arbitrary way in time. An analytical expression for the wave function in arbitrary instant time was built. It is shown the dynamic shift of the center of the potential doesn’t change the variance. Moreover, the system can exhibit the resonance: when the frequency of the potential perturbation approaches to the natural frequency the amplitude of the wave packet of particle is increased.