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arXiv:1311.2351

Paradoxical Quantum Scattering off a Time Dependent Potential?

Ori Reinhardt, Moshe Schwartz

Published Nov 11, 2013• 0 citations
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Abstract

We consider the quantum scattering off a time dependent barrier in one dimension. Our initial state is a right going eigenstate of the Hamiltonian at time t=0. It consists of a plane wave incoming from the left, a reflected plane wave on the left of the barrier and a transmitted wave on its right. We find that at later times, the evolving wave function has a finite overlap with left going eigenstates of the Hamiltonian at time t=0. For simplicity we present an exact result for a time dependent delta function potential. Then we show that our result is not an artifact of that specific choice of the potential. This surprising result does not agree with our interpretation of the eigenstates of the Hamiltonian at time t=0. A numerical study of evolving wave packets, does not find any corresponding real effect. Namely, we do not see on the right hand side of the barrier any evidence for a left going packet. Our conclusion is thus that the intriguing result mentioned above is intriguing only due to the semantics of the interpretation.

DOI
10.48550/arxiv.1311.2351

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We consider the quantum scattering off a time dependent barrier in one dimension. Our initial state is a right going eigenstate of the Hamiltonian at time t=0. It consists of a plane wave incoming from the left, a reflected plane wave on the left of the barrier and a transmitted wave on its right. We find that at later times, the evolving wave function has a finite overlap with left going eigenstates of the Hamiltonian at time t=0. For simplicity we present an exact result for a time dependent delta function potential. Then we show that our result is not an artifact of that specific choice of the potential. This surprising result does not agree with our interpretation of the eigenstates of the Hamiltonian at time t=0. A numerical study of evolving wave packets, does not find any corresponding real effect. Namely, we do not see on the right hand side of the barrier any evidence for a left going packet. Our conclusion is thus that the intriguing result mentioned above is intriguing only due to the semantics of the interpretation.
The time evolution of the probability and current densities stemming from an initial state of a particle in the form of a wave packet on the left hand side of a square barrier is studied. In particular, the space distribution at a given time of the above densities is given. Furthermore, the evolution in time at the entrance and exit of the barrier is given. The numerical results show repeated reversal in the current density at the barrier entrance, while being unidirectional at the exit. However, the probability entering the barrier over a long time equals the corresponding probability exiting the barrier. Owing to the fact that the wave packet expands on both of its sides, as time goes by, it is possible to have transmission even if the particle’s initial momentum points away from the barrier. The effect, in question, becomes evident in a diagram for the transmission coefficient in terms of the particle initial momentum, over a relatively small region.
In this work, the motion of a particle in one dimension under the influence of the linearly downward potential well is studied within the context of the non-relativistic quantum mechanics. The attention is paid on the paradoxical phenomenon of the reflection of a particle that is in contrast between classical and quantum physics. Classically, the reflection effect occurs only at a potential barrier. To demonstrate such counter-intuitive phenomenon, the Schrödinger equation is solved to obtain the reflection coefficient in the scattering state by considering an incident particle that is represented by a monochromatic plane wave having an energy E > 0, propagates freely from left to right, pass through the potential well. The continuity conditions at boundaries give the desired result that is expressed in terms of the Airy functions which depends on the incident energy E, the strength jV 0 j and the range L of the well. The value of the reflection coefficient R lies in the interval 0 < R < 1, and its behavior is the decreasing function with respect to the range L.
Scattering a particle on a one-dimensional δ\deltaδ-potential barrier: asymptotic superselection rule
As is stated, the modern mathematical (C∗C^*C∗-algebraic) scattering theory for the formal Hamiltonian with a one-dimensional short-range potential, developed on the basis of the operational definition of observables, describes unitary asymptotically free dynamics. But, as it s shown in this paper by the example of the δ\deltaδ-potential, this Hamiltonian can describe either unitary asymptotically not free dynamics (endless interaction process) or non-unitary asymptotically free dynamics (scattering process). In the case of scattering, the unboundedness of the position operator plays a key role. Each solution to the Schrodinger equation, describing the scattering process with a one-sided incidence of a particle on the barrier, is non-unique in the limit t→∞t\to \inftyt→∞, when it is a superposition of unconnected states (left and right out-asymptotes) localized in the disjoint spatial regions located on opposite sides of the barrier --- it describes non-unitary quantum dynamics. Measurement of the coordinates of the transmitted and reflected particles using one experimental setup of finite dimensions, as is assumed in the operational approach, is impossible. We need two such setups --- one for the transmitted particles, and the other for the reflected ones. Thus, such a solution is a mixed vector state --- this process is governed by the asymptotic superselection rule.
We consider the time development of a one-dimensional Gaussian wave packet. Noting that the quantum mechanical predictions concerning position and momentum measurements of a free particle described by a Gaussian packet are consistent with the predictions one would make on the basis of a classical “Naive Model,” we investigate whether this model can account for the scattering of a particle by a rectangular barrier. Obtaining numerical solutions of the time dependent Schrödinger equation for a series of packets with varying incident momenta, we find that the Naive Model is inadequate to explain the quantum mechanical transmission. We study the relative speeds of the transmitted and reflected packets, discussing a number of possible interpretations and indicating the quantum mechanical result. Finally, we note that for certain choices of the parameters describing the packet and the barrier, the reflected packet is composed of two distinct humps.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.