A quantum particle in a box with moving walls (original) (raw)
International Journal of Theoretical Physics, 2011
In this paper I present a mapping between representation of some quantum phenomena in one dimension and behavior of a classical time-dependent harmonic oscillator. For the first time, it is demonstrated that quantum tunneling can be described in terms of classical physics without invoking violations of the energy conservation law at any time instance. A formula is presented that generates a wide class of potential barrier shapes with the desirable reflection (transmission) coefficient and transmission phase shift along with the corresponding exact solutions of the time-independent Schrödinger's equation. These results, with support from numerical simulations, strongly suggest that two uncoupled classical harmonic oscillators, which initially have a 90° relative phase shift and then are simultaneously disturbed by the same parametric perturbation of a finite duration, manifest behavior which can be mapped to that of a single quantum particle, with classical 'range relations' analogous to the uncertainty relations of quantum physics.
Dynamics of a particle confined in a two- or three-dimensional moving domain
The behavior of a quantum particle confined in a two-dimensional box whose walls are moving is investigated and the relevant mathematical problem with moving boundaries is recast in the form of a problem with fixed boundaries and time-dependent Hamiltonian. Changes of the shape of the box are shown to be important, as it clearly emerges from the comparison between the 'pantographic' case (same shape of the box through all the process) and the case with deformation. Extension of the results to the three-dimensional case is also briefly discussed.
Untouched aspects of the wave mechanics of a particle in one dimensional box
2006
Wave mechanics of a particle in 1-D box (size = d) is critically analyzed to reveal its untouched aspects. When the particle rests in its ground state, its zero-point force (F o) produces non-zero strain by modifying the box size from d to d ′ = d + ∆d in all practical situations where the force (F a) restoring d is not infinitely strong. Assuming that F a originates from a potential ∝ x 2 (x being a small change in d), we find that: (i) the particle and strained box assume a mutually bound state (under the equilibrium between F o and F a) with binding energy ∆E = −ε ′ o ∆d/d ′ (with ε ′ o = h 2 /8md ′2 being the ground state energy of the particle in the strained box), (ii) the box size oscillates around d ′ when the said equilibrium is disturbed, (iii) an exchange of energy between the particle and the strained box occurs during such oscillations, and (iv) the particle, having collisional motion in its excited states, assumes collisionless motion in its ground state. These aspects have desired experimental support and proven relevance for understanding the physics of widely different systems such as quantum dots, quantum wires, trapped single particle/ion, clusters of particles, superconductors, superfluids, etc. It is emphasized that the physics of such a system in its low energy states can be truly revealed if the theory incorporates F o and related aspects.
Classical path from quantum motion for a particle in a transparent box
We consider the problem of a free particle inside a one-dimensional box with transparent walls (or equivalently, along a circle with a constant speed) and discuss the classical and quantum descriptions of the problem. After calculating the mean value of the position operator in a time-dependent normalized complex general state and the Fourier series of the function position, we explicitly prove that these two quantities are in accordance by (essentially) imposing the approximation of high principal quantum numbers on the mean value. The presentation is accessible to advanced undergraduate students with a knowledge of the basic ideas of quantum mechanics.
Quantum dynamics of a hydrogen-like atom in a time-dependent box: non-adiabatic regime
The European Physical Journal D, 2018
We consider a hydrogen atom confined in time-dependent trap created by a spherical impenetrable box with time-dependent radius. For such model we study the behavior of atomic electron under the (non-adiabatic)dynamical confinement caused by the rapidly moving wall of the box. The expectation values of the total and kinetic energy, average force, pressure and coordinate are analyzed as a function of time for linearly expanding, contracting and harmonically breathing boxes. It is shown that linearly extending box leads to de-excitation of the atom, while the rapidly contracting box causes the creation of very high pressure on the atom and transition of the atomic electron into the unbound state. In harmonically breathing box diffusive excitation of atomic electron may occur in analogy with that for atom in a microwave field.
Quantum systems with time-dependent boundaries
International Journal of Geometric Methods in Modern Physics, 2015
We present here a set of lecture notes on quantum systems with time-dependent boundaries. In particular, we analyze the dynamics of a non-relativistic particle in a bounded domain of physical space, when the boundaries are moving or changing. In all cases, unitarity is preserved and the change of boundaries does not introduce any decoherence in the system.
Quantum dynamics of PT-symmetric kicked particle in a 1D box
Journal of Physics A: Mathematical and Theoretical
We study quantum particle dynamics in a box and driven by PT-symmetric, delta-kicking complex potential. Such dynamical characteristics as the average kinetic energy as function of time and quasienergy at different values of the kicking parameters. Breaking of the PT-symmetry at certain values of the non-Hermitian kicking parameter is shown. Experimental realization of the model is also discussed.
Quantum particle displacement by a moving localized potential trap
EPL (Europhysics Letters), 2009
We describe the dynamics of a bound state of an attractive δ-well under displacement of the potential. Exact analytical results are presented for the suddenly moved potential. Since this is a quantum system, only a fraction of the initially confined wavefunction remains confined to the moving potential. However, it is shown that besides the probability to remain confined to the moving barrier and the probability to remain in the initial position, there is also a certain probability for the particle to move at double speed. A quasi-classical interpretation for this effect is suggested. The temporal and spectral dynamics of each one of the scenarios is investigated.
The particle in a box in PT quantum mechanics and an electromagnetic analog
2017
In PT quantum mechanics a fundamental principle of quantum mechanics, that the Hamiltonian must be hermitian, is replaced by another set of requirements, including notably symmetry under PT, where P denotes parity and T denotes time reversal. Here we study the role of boundary conditions in PT quantum mechanics by constructing a simple model that is the PT symmetric analog of a particle in a box. The model has the usual particle in a box Hamiltonian but boundary conditions that respect PT symmetry rather than hermiticity. We find that for a broad class of PT-symmetric boundary conditions the model respects the condition of unbroken PT-symmetry, namely that the Hamiltonian and the symmetry operator PT have simultaneous eigenfunctions, implying that the energy eigenvalues are real. We also find that the Hamiltonian is self-adjoint under the PT inner product. Thus we obtain a simple soluble model that fulfils all the requirements of PT quantum mechanics. In the second part of this pape...