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Manipulation of quantum evolution

David Jose Fernandez Cabera, Bogdan Mielnik

Published May 1, 1994• 1 citation
OpenAlex

Abstract

The free evolution of a non-relativistic charged particle is manipulated using time-dependent magnetic fields. It is shown that the application of a programmed sequence of magnetic pulses can invert the free evolution process, forcing an arbitrary wave packet to 'go back in time' to recover its past shape. The possibility of more general operations upon the Schrodinger wave packet is discussed.

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Relativistic Dynamics, Electromagnetic Field, and Spin, as All Quantum Effects

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Time-development operator method in quantum mechanics

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We discuss the time-development operator method in quantum mechanics. The equivalence of this method and the usual method of expansion in terms of energy eigenfunctions discussed in textbooks is pointed out. As examples of cases of time-dependent Hamiltonians, we discuss the time development of a Gaussian wave packet for a charged particle subject to a time-dependent electric field using an operator differential equation. We also consider a spin in a time-dependent magnetic field through a time-development operator.

American Journal of Physics•Apr 1, 2001•13 citations
The free evolution of a non-relativistic charged particle is manipulated using time-dependent magnetic fields. It is shown that the application of a programmed sequence of magnetic pulses can invert the free evolution process, forcing an arbitrary wave packet to 'go back in time' to recover its past shape. The possibility of more general operations upon the Schrodinger wave packet is discussed.
The techniques of controlling the free evolution of a nonrelativistic charged particle by time-dependent magnetic fields are proposed and the possibility of more general operations upon the Schrödinger wave packet is discussed. It is found that a properly programmed sequence of magnetic pulses can invert the free evolution process, forcing an arbitrary wave packet to ‘‘go back in time’’ to recover its past shape. Our manipulation prescriptions hold also for nonrelativistic particles of arbitrary spin.
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.
It has been shown that the irreversible dynamics of a system in a dissipative environment can be described by an effective one-particle Hamiltonian (in the classical case) or an effective one-particle Schrödinger equation (in the quantum mechanical case) without taking into account the environmental degrees of freedom explicitly. In the quantum mechanical case, the description can be achieved either by a nonlinear modification of the Schrödinger equation or an explicitly time-dependent Hamiltonian operator — where both forms are connected via a non-unitary transformation, corresponding to a non-canonical transformation on the classical level. Taking into account an external magnetic field, in both formalisms, an additional unphysical gauge-dependent term occurs in the equations of motion. This shortcoming can be eliminated if the Lagrangian and Hamiltonian that provide Maxwell’s equations of electrodynamics undergo, in the dissipative case, the same kind of transformation as the mechanical degrees of freedom of the system. As a result, a consistent description in terms of exact analytical wave packet-solutions for the dissipative motion in a magnetic field can be obtained that will be discussed in detail.
We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe the particle dynamics, are in agreement with the Hamilton equations only if in the time dependent phases one considers the Lagrangian instead of the Hamiltonian which leads to the conventional Schrodinger equation. We define a relativistic quantum principle asserting that a quantum particle has a finite frequency spectrum, with a cutoff propagation velocity c as a universal constant not depending on the coordinate system, and that any time dependent phase variation is the same in any system of coordinates. From the time dependent phase invariance, the relativistic kinematics is obtained. We consider two types of possible interactions: 1) An interaction with an external field, by a modification of the time dependent phase differential with the terms proportional to the differentials of the space-time coordinates multiplied with the components of this field four-potential, and 2) an interaction by a deformation of the space-time coordinates, due to a gravitational field. From the invariance of the time dependent phase with field components, we obtain a mechanical force of the form of Lorentz’s force, and three Maxwell equations: The Gauss-Maxwell equations for the electric and magnetic fluxes, and the Faraday-Maxwell equation for the electromagnetic induction. When the fourth equation, Ampere-Maxwell, is considered, the interaction field takes the form of the electromagnetic field. For a low propagation velocity of the particle waves, we get a packet of waves with the time dependent phases proportional to the relativistic Hamiltonian, as in Dirac’s famous theory of spin, and a slowly-varying amplitude with a phase proportional to the momentum and this velocity. In the framework of our theory, the spin is obtained as an all quantum effect, without any additional assumption to the quantum theory. When a space-time deformation is considered in the time dependent phase of a quantum particle, from the group velocity we get the particle dynamics according to the general theory of relativity. In this way, the relativistic dynamics, the electromagnetic field, and the spin of a quantum particle are obtained only from the invariance of the time dependent phases of the particle wave functions.
We discuss the time-development operator method in quantum mechanics. The equivalence of this method and the usual method of expansion in terms of energy eigenfunctions discussed in textbooks is pointed out. As examples of cases of time-dependent Hamiltonians, we discuss the time development of a Gaussian wave packet for a charged particle subject to a time-dependent electric field using an operator differential equation. We also consider a spin in a time-dependent magnetic field through a time-development operator.