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

ABSTRACT

The mean square displacement ${〈{\left(x\left(t\right)-x\left(0\right)\right)}^2〉}_{}$ 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 $\hslash /m$, 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.

GRAPHICAL ABSTRACT

KEYWORDS:

• Quantum dynamics
• space–time pair correlation function
• mean square displacement
• surface diffusion
• Brownian motion

Acknowledgments

The author thanks Peter Saalfrank for bringing to his attention the question about the quantum mechanical time evolution of the MSD; very helpful discussions with him as well as with Salvador Miret-Artés, Jörn Manz, Fabien Gatti, Jean-Christophe Tremblay and Emmanuel Fromager are gratefully acknowledged. The author dedicates this paper to Prof. Jürgen Troe on the occasion of his 80th birthday.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 According to Resolution 1 of the 26th Conference of Weights and Measures (Comptes Rendus des séances de la vingt-sixième Conférence Générale des Poids et Mesures. Résolution 1, Annexe 3, BIPM, 2019), as of 20 Mai 2019 the Boltzmann and the Planck constants have the fixed values $k_{\mathrm{B}}=1.380649{10}^{-23}\mathrm{J}/\mathrm{K}$ and $h=6.62607015{10}^{-34}\mathrm{J}\mathrm{s}$, respectively.

Additional information

Funding

This work was supported by Agence Nationale de la Recherche [project QDDA].