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Quantum-mechanical time evolution and uniform forces

Gary Bowman

Published Oct 1, 2006• 0 citations
OpenAlex

Abstract

The time evolution of an arbitrary one-dimensional initial quantum state, subject to a time-independent, spatially uniform classical force, is shown to be that of a free-particle state, plus an overall motion arising from the classical force. Bohmian mechanics is then used to extend this result to an arbitrarily time-dependent, uniform force in three dimensions. The resulting solution is completed and confirmed using conventional quantum mechanics.

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arXiv:1402.0396•European Journal of Physics•May 9, 2014•4 citations
The time evolution of an arbitrary one-dimensional initial quantum state, subject to a time-independent, spatially uniform classical force, is shown to be that of a free-particle state, plus an overall motion arising from the classical force. Bohmian mechanics is then used to extend this result to an arbitrarily time-dependent, uniform force in three dimensions. The resulting solution is completed and confirmed using conventional quantum mechanics.
We present a simple approach for finding the general solution of the time-dependent Schrödinger equation for a particle moving under the action of a uniform force. From such a general solution we derive the eigenfunctions of the energy as well as those of the momentum. Next we obtain the propagator and show that the time evolution of an arbitrarily chosen initial wavefunction is described by a Fresnel transformation. This establishes a useful link with optical phenomena and allows us to link the case of a particle moving under a uniform force to that of the free particle.
Trajectories of a Bohmian particle confined in time-dependent cylindrical and spherical traps are computed for both contracting and expanding boxes. Quantum effective force is considered in arbitrary directions. It is seen that in contrast to the problem of a particle in an infinite rectangular box with one wall in motion, if particle initially is in an energy eigenstate of a tiny box the force is zero in all directions. Trajectories of a two-body system confined in the spherical trap are also computed for different statistics. Computations show that there are situations for which distance between bosons is greater than the fermions. However, results of average separation of the particles confirm our expectation about the statistics.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.