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Molecular Physics

The mean square displacement of a ballistic quantum particle

Oussama Bindech, Fabien Gatti, Souvik Mandal, Roberto R. Marquardt, Lei Shi, Jean Christophe Tremblay

Published Mar 4, 2024• 1 citation

Abstract

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.

DOI: 10.1080/00268976.2024.2322023

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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 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.

It is well-accepted that the Dirac delta function can be defined by the limitation of the square function covering a unit area with decrease of the width of the square to zero, and the eigenvectors of the position operator take the form of the delta function. When discussing the position measurement in quantum mechanics, one is prompted by the mathematical convention that uses the square function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large. The continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the delta function to represent the physical states of position.

In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.

Recently Costa Filho et al. [Phys. Rev. A 84, 050102(R) (2011)] have introduced a position-dependent infinitesimal translation operator which corresponds to a position-dependent linear momentum and consequently to a quantum particle with position-dependent effective mass. Although there is no doubt about the novelty of the idea and the formalism, we believe that some aspects of the quantum mechanics in their original work could be enhanced. Here in this Brief Report first we address those points and then an alternative is introduced. Finally we apply the formalism for a quantum particle under a null potential confined in a square well, and the results are compared with those in the paper mentioned above.

The Dirac delta function can be defined by the limitation of the rectangular function covering a unit area with decrease of the width of the rectangle to zero, and in quantum mechanics the eigenvectors of the position operator assume the form of the delta function. When discussing the position measurement in quantum mechanics, one is tempting to take the mathematical convention that uses the rectangular wave function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large. The continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the delta function to represent the physical states of the position. When the slit experiment is taken as an apparatus of position measurement, no matter what potential is used to model the slit, only the ground state of the slit-dependent wave function matters.

A quantum-mechanical analysis is made of the experimental accuracy to be expected for particle-counting and intensity-correlation experiments. The mean-square fluctuation for an ensemble, consisting of a large number of experiments each conducted over a time interval

T

, is calculated.