Conditions for the classicality of the center of mass of many-particle quantum states

Abstract

We discuss the conditions for the classicality of quantum states with a very large number of identical particles. By defining the center of mass from a large set of Bohmian particles, we show that it follows a classical trajectory when the distribution of the Bohmian particle positions in a single experiment is always equal to the marginal distribution of the quantum state in physical space. This result can also be interpreted as a single experiment generalization of the well-known Ehrenfest theorem. We also demonstrate that the classical trajectory of the center of mass is fully compatible with a quantum (conditional) wave function solution of a classical non-linear Schrödinger equation. Our work shows clear evidence for a quantum-classical inter-theory unification, and opens new possibilities for practical quantum computations with decoherence.found. This is the many worlds solution [18][19][20], in which the famous Schrödinger's cat is found alive in one world and dead in another. Explanations of the quantum-to-classical transition have also been attempted within this interpretation [20].There is a final kind of solution that assumes that the wave function alone does not provide a complete description of the quantum state, i.e. additional elements (hidden variables) are needed. The most widespread of these approaches is Bohmian mechanics [10,[23][24][25][26][27][28], where, in addition to the wave function, well-defined trajectories are needed to define a complete (Bohmian) quantum state. In a spatial superposition of two disjoint states, only the one whose support contains the position of the particle becomes relevant for the dynamics. Previous attempts to study the quantum-to-classical transition with Bohmian mechanics mainly focused on single-particle problems [28][29][30][31]. In this paper, we generalize such works by analyzing under which conditions the center of mass of a many-particle quantum system follows a classical trajectory.The use of the center of mass for establishing the classicality of a quantum state has some promising advantages. The first one is related to the description of the initial conditions. Fixing the initial position and velocity of a classical particle seems unproblematic, while it is forbidden for a quantum particle due to the uncertainty principle [1,14]. The use of the center of mass relaxes this contradiction: it is reasonable to expect that two experiments with the same preparation for the wave function will give quite similar values for the initial position and velocity of the center of mass when a large number of particles is considered, although the microscopic distribution of the positions and velocities for all (Bohmian) particles will be quite different in each experiment.The second advantage is that it provides a natural coarse-grained definition of a classical trajectory that coexists with the underlying microscopic quantum reality. One can reasonably expect that the Bohmian trajectory of the center of mass of a large number of particles can follow a classical trajec...

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