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

Abstract

The mean square displacement $⟨{\left(x(t)-x(0)\right)}^2⟩$ 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.

21 pages, 2 figures, 59 references, 5 pages appendix

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