Classical and quantum dynamics of a free particle in a driven potential inside a rigid box

M. G. E. da Luz, Bin Kang Cheng

The semiclassical approximation has been used to evaluate the exact propagators for two one-dimensional quantum systems: (a) a free particle interacting with one hard-wall potential moving with constant velocity, and (b) a free particle inside a rigid box with one wall moving uniformly in time. The results derived from these propagators are in agreement with those of the corresponding Schrodinger equations.

Claude Aslangul, No�lle Pottier, Daniel Saint-James

Jambul Yusupov, D.M. Otajanov, V. E. Eshniyazov +1 more

Katsunori Mita

Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β2=ℏ/2m. In this article, we explore the implications of this view. Rewriting the wave function in a polar form and transforming Schrödinger's equation into two real equations, we show that one of them reduces to the continuity equation, and the other, a nonlinear dynamical equation for the probability density. Considering these two equations as if they were the basic equations of quantum mechanics, we apply them to several one-dimensional quantum systems. We show the dispersive properties in the probability densities of stationary states of a particle in a rigid box and in harmonic potential; quasi-classical Gaussian probability densities of a free particle; and coherent states and squeezed states of the harmonic oscillator. We also present the soliton as a quantum mechanical representation of a free particle. We discuss the meaning of the diffusion coefficient β2 for each quantum system using a density plot.

Leal Edgar

The perturbative response of the hydrogen atom to a constant external electric field (Linear Stark effect) is analyzed. While a classical planetary model (based on the conservation of angular momentum ) predicts an amplification of the Coulomb tension by a factor of 2, the exact quantum solution yields a factor of 2/3. We analytically demonstrate that this 2/3 factor is identical to that obtained in a classical system under a potential subjected to a rigid angular frequency restriction (constant ). This result suggests that the degenerate manifold of hydrogen behaves dynamically not as a Keplerian free particle, but as an internally driven coherent oscillator, where the effective orbital frequency is quantized and protected against adiabatic radial perturbations. We discuss the implications of this rigidity on the polarizability of Rydberg states and its application in quantum technologies.

Zoran Hadzibabic