From Ehrenfest to Schrödinger

Klaus Renziehausen, Ingo Barth

Applying the concept of quantum hydrodynamics on a hydrogen atom, one can derive an equation of motion which describes the time derivative of the mass current density for the relative motion of the electron and the proton. Since this derivation can be performed with the Ehrenfest theorem, we call this equation the Ehrenfest equation of motion. As quantum hydrodynamics is consistent with Schrödinger's quantum mechanics, the stationary s-wave functions satisfy both the Schrödinger equation and the Ehrenfest equation of motion. It is the purpose of this paper to prove that the stationary s-wave functions satisfy the Ehrenfest equation of motion. From the quantum hydrodynamical point of view, this result for s-wave functions can be interpreted that the Coulomb force density that attracts the electron to the proton is compensated by a quantum force density that is related to the dispersion of the probability density.

G Kälbermann

We prove that Galilean invariant Schrödinger equations derived from Lagrangian densities necessarily obey the Ehrenfest theorem for velocity-independent potentials. The conclusion holds as well for Lagrangians describing nonlinear self-interactions. An example of Doebner and Goldin motivates the result.

Léopold Halpern

Ijaz Durrani

J L Alonso, A Castro, Jesús Clemente-Gallardo +3 more

Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.