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arXiv:2305.03847 · Europhysics Letters (EPL)

Evolution of the wave function's shape in a time-dependent harmonic potential

Etera R. Livine

Published Aug 1, 2023• 2 citations
arXivHTMLPDF

Abstract

Abstract An effective operational approach to quantum mechanics is to focus on the evolution of wave packets, for which the wave function can be seen in the semi-classical regime as representing a classical motion dressed with extra degrees of freedom describing the shape of the wave packet and its fluctuations. These quantum dressing are independent degrees of freedom, mathematically encoded in the higher moments of the wave function. We review how to extract the effective dynamics for Gaussian wave packets evolving according to the Schrödinger equation with time-dependent potential in a 1 + 1-dimensional spacetime, and derive the equations of motion for the quadratic uncertainty. We then show how to integrate the evolution of all the higher moments for a general wave function in a time-dependent harmonic potential.

DOI
10.1209/0295-5075/acedcc

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Abstract An effective operational approach to quantum mechanics is to focus on the evolution of wave packets, for which the wave function can be seen in the semi-classical regime as representing a classical motion dressed with extra degrees of freedom describing the shape of the wave packet and its fluctuations. These quantum dressing are independent degrees of freedom, mathematically encoded in the higher moments of the wave function. We review how to extract the effective dynamics for Gaussian wave packets evolving according to the Schrödinger equation with time-dependent potential in a 1 + 1-dimensional spacetime, and derive the equations of motion for the quadratic uncertainty. We then show how to integrate the evolution of all the higher moments for a general wave function in a time-dependent harmonic potential.
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.
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.
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.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
We discuss nonspreading wave packets in one dimensional Schrödinger equation. We derive general rules for constructing nonspreading wave packets from a general potential V(x,t)\textmd{V}(x,t)V(x,t). The essential ingredients of a nonspreading wave packet, the shape function f(x)f(x)f(x), the motion d(t)d(t)d(t), the phase function ϕ(x,t)ϕ(x,t)ϕ(x,t) are derived. Since the form of the shape of a nonspreading wave packet does not change in time, the shape equation should be time independent. We show that the shape function f(x)f(x)f(x) is the eigenfunction of the time independent Schrödinger equation with an effective potential VeffV_{\textmd{eff}}Veff​ and an energy EeffE_{\textmd{eff}}Eeff​. We derive nonspreading wave packets found by Schrödinger, Senitzky, and Berry and Balazs as examples. We show that most stationary potentials can only support stationary nonspreading wave packets. We show how to construct moving nonspreading wave packets from time dependent potentials, which drive nonspreading wave packets into an arbitrary motion.
The evolution of the center-of-mass wave function for a mesoscopic particle according to the Schr\"odinger-Newton equation can be approximated by a harmonic potential if the wave function is narrow compared to the size of the mesoscopic particle. It was noticed by Colin et al. [Phys. Rev. A 93, 062102 (2016).] that, in the regime where self-gravitational effects are weak, intermediate and wider wave functions may be approximated by a harmonic potential as well but with a width-dependent coupling, leading to a time evolution that is determined only by a differential equation for the width of a Gaussian wave function as a single parameter. Such an approximation results in considerably less computational effort in order to predict the self-gravitational effects on the wave-function dynamics. Here, we provide an alternative approach to this kind of approximation, including a rigorous derivation of the equations of motion for an initially Gaussian wave packet under the assumption that its shape is conserved. Our result deviates to some degree from the result by Colin et al. [Phys. Rev. A 93, 062102 (2016).], specifically in the limit of wide wave functions.