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Mathematical Proceedings of the Cambridge Philosophical Society

On the motion of a Gaussian wave-packet in a parabolic potential field

Charles Alfred Coulson, George Stanley Rushbrooke

Published Oct 1, 1946• 2 citations
Publisher

Abstract

The paper deals with the motion of a Gaussian wave-packet in a parabolic potential field, contrasting the quantum-mechanical behaviour of the packet with that which follows from pure classical mechanics. The behaviour is entirely classical only if the initial phase of the space-wave-function is linear in the space coordinate; in which case the wave-packet is initially a ‘best-possible’ packet according to Heisenberg's uncertainty relationship. The paper concludes with a brief reference to the similar problem for a free particle. Note added in Proof, 16 February 1946. Through the courtesy of Mr M. S. Bartlett, we have been privileged to see the manuscripts of some work by Mr J. E. Moyal and himself, in which the motion of a probability wave-packet is discussed in more general terms than our own: and in which it appears that the motion of our wave-packet, which we have described as non-classical, is indeed actually classical provided that a certain correlation exists between momentum and position in the p, q-distribution function. With this knowledge we now find that classical and quantal predictions agree precisely provided that the initial classical probability function is given by this is a particular case of their more general results.

DOI
10.1017/s0305004100023057

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The paper deals with the motion of a Gaussian wave-packet in a parabolic potential field, contrasting the quantum-mechanical behaviour of the packet with that which follows from pure classical mechanics. The behaviour is entirely classical only if the initial phase of the space-wave-function is linear in the space coordinate; in which case the wave-packet is initially a ‘best-possible’ packet according to Heisenberg's uncertainty relationship. The paper concludes with a brief reference to the similar problem for a free particle. Note added in Proof, 16 February 1946. Through the courtesy of Mr M. S. Bartlett, we have been privileged to see the manuscripts of some work by Mr J. E. Moyal and himself, in which the motion of a probability wave-packet is discussed in more general terms than our own: and in which it appears that the motion of our wave-packet, which we have described as non-classical, is indeed actually classical provided that a certain correlation exists between momentum and position in the p, q-distribution function. With this knowledge we now find that classical and quantal predictions agree precisely provided that the initial classical probability function is given by this is a particular case of their more general results.
Abstract In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrödinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.
By modelling quantum systems as emerging from a (classical) sub-quantum thermodynamics, the quantum mechanical "decay of the wave packet" is shown to simply result from sub-quantum diffusion with a specific diffusion coefficient varying in time due to a particle's changing thermal environment. It is thereby proven that free quantum motion strictly equals ballistic diffusion. The exact quantum mechanical trajectory distributions and the velocity field of the Gaussian wave packet are thus derived solely from classical physics. Moreover, also quantum motion in a linear (e.g., gravitational) potential is shown to equal said ballistic diffusion. Quantitative statements on the trajectories' characteristic behaviours are obtained which provide a detailed "micro-causal" explanation in full accordance with momentum conservation.
In this paper,the propagation nature of the wave packet was discussed through the Green's function under three common potential fields.We had discovered some valuable conclusions.Firstly,when the wave packet of free particle varied with the time,the wave packet center of gravity moved by a speed 0/μ, but when the wave packet amplitude changed from the initial δ to the final δt,the Position variable r still deferred to the Gaussian distribution. Secondly,the wave packet amplitude in a homogeneous field changed by a power series with relative to the initial time.Finally,in simple harmonics oscillator potential field,the momentum mean value in random time is precisely the momentum form of classical simple harmonics oscillator.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
Variational Gaussian wave-packet dynamics is usually obtained by restricting quantum evolution to Gaussian wave packets. We instead construct finite-NNN dynamics for a labeled bundle of classical trajectories under a constraint fixing the symplectic-area scale of its covariance. At finite NNN, the bundle itself is the dynamical state: it may be non-Gaussian, and the potential force on each trajectory is evaluated from the full potential without a local harmonic approximation. For sequences whose initial empirical distributions approach a smooth Gaussian density as N→∞N \to \inftyN→∞, the limiting relative motion becomes linear and preserves Gaussianity. With the area scale set to ℏ/2\hbar/2ℏ/2, the limiting centroid and width equations coincide with those of the Gaussian time-dependent variational principle, and the limiting phase-space density coincides with the corresponding Gaussian Wigner density for matched initial data. Here NNN counts trajectories, not orders in a Moyal or ℏ\hbarℏ expansion. This phase-space correspondence reveals two complementary constructions of the same Gaussian dynamics: a variational reduction of quantum evolution and the Gaussian large-NNN limit of a constrained classical trajectory bundle.
A motion of a Gaussian wave packet of a single electron in a harmonic potential is obtained by solving the time dependent Schrödinger equation analytically. The analytic formula of the phase of the wave function which depends on both time and space is also presented explicitly. The probability density of the wave packet moves changing its width and central position periodically. The period of the change of the width is half of that of the central position. The interpretation of these periodic changes in terms of the Green's function which can be derived by the path integration formulation is also discussed.