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The Journal of Chemical Physics

Wave packet dynamics in an harmonic potential disturbed by disorder: Entropy, uncertainty, and vibrational revivals

Peter Schürger, Thomas Schaupp, Dustin Kaiser, Bernd Engels, Volker Engel

Published Jan 13, 2022• 11 citations
Publisher

Abstract

We investigate the quantum and classical wave packet dynamics in an harmonic oscillator that is perturbed by a disorder potential. This perturbation causes the dispersion of a Gaussian wave packet, which is reflected in the coordinate-space and the momentum-space Shannon entropies, the latter being a measure for the amount of information available on a system. Regarding the sum of the two quantities, one arrives at an entropy that is related to the coordinate-momentum uncertainty. Whereas in the harmonic case, this entropy is strictly periodic and can be evaluated analytically, this behavior is lost if disorder is added. There, at selected times, the quantum mechanical probability density resembles that of a classical oscillator distribution function, and the entropy assumes larger values. However, at later times and dependent on the degree of disorder and the chosen initial conditions, quantum mechanical revivals occur. Then, the observed effects are reversed, and the entropy may decrease close to its initial value. This effect cannot be found classically.

DOI
10.1063/5.0079938

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We investigate the quantum and classical wave packet dynamics in an harmonic oscillator that is perturbed by a disorder potential. This perturbation causes the dispersion of a Gaussian wave packet, which is reflected in the coordinate-space and the momentum-space Shannon entropies, the latter being a measure for the amount of information available on a system. Regarding the sum of the two quantities, one arrives at an entropy that is related to the coordinate-momentum uncertainty. Whereas in the harmonic case, this entropy is strictly periodic and can be evaluated analytically, this behavior is lost if disorder is added. There, at selected times, the quantum mechanical probability density resembles that of a classical oscillator distribution function, and the entropy assumes larger values. However, at later times and dependent on the degree of disorder and the chosen initial conditions, quantum mechanical revivals occur. Then, the observed effects are reversed, and the entropy may decrease close to its initial value. This effect cannot be found classically.
If the probability density P ( x , t ) = |ψ ( x , t )| 2 of a non-stationary state happens to meet the condition P ( x , t 0 ) = P ( x , t 0 + Υ), the wave packet is said to experience exact revivals at intervals of Υ. The definition implies that a harmonic oscillator wave packet will experience revivals with a period Υ = 2π/ω, where ω is the angular frequency of the corresponding classical system. A harmonic oscillator wave packet of Gaussian form, centered initially ( t = 0) at x = x 0 , is shown to spread almost everywhere, if its initial width is sufficiently narrow, in a time very much shorter than the period of the corresponding classical oscillator. However, the packet reassembles itself twice during a single period: (i) it experiences a mirror revival, defined by the relation P ( x , ½Υ) = P (- x , 0), and (ii) an exact revival, P ( x , Υ) = P ( x , 0). These results are used to counter a recent claim (which apparently gives voice to a commonly held belief) that a harmonic oscillator wave packet ψ( x , t ) does not collapse and has no revivals. Plots of the recurrence probability |⟨ψ( x , 0)|ψ( x , t )⟩| 2 are presented and the need for formulating unambiguous definitions of the two concepts, revival and collapse, is stressed.
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.
This work is a simple example of the quantum dynamics of a particle in a disordered system in one dimension.In particular, we illustrate numerically the phenomenon of Anderson localisation of a wave packet using a basic model constructed with small random rectangular potential barriers.Also, we study the dynamics of a quantum particle in a disordered potential formed by an harmonic oscillator perturbed by random rectangular barriers.To show the effects of disorder on the dynamics of the system, we compare the time evolution of the wave function of the unperturbed harmonic oscillator with the wave function of the disordered system.We do this by taking the scalar product between the unperturbed and perturbed wave functions at each timestep for different values of the perturbation parameters affecting the disordered wave packet.
The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.
The autocorrelation function, A(t), measures the overlap (in Hilbert space) of a time-dependent quantum mechanical wave function, psi(x,t), with its initial value, psi(x,0). It finds extensive use in the theoretical analysis and experimental measurement of such phenomena as quantum wave packet revivals. We evaluate explicit expressions for the autocorrelation function for time-dependent Gaussian solutions of the Schrodinger equation corresponding to the cases of a free particle, a particle undergoing uniform acceleration, a particle in a harmonic oscillator potential, and a system corresponding to an unstable equilibrium (the so-called `inverted' oscillator.) We emphasize the importance of momentum-space methods where such calculations are often more straightforwardly realized, as well as stressing their role in providing complementary information to results obtained using position-space wavefunctions.
We investigate the short-, medium-, and long-term time dependence of wave packets in the infinite square well. In addition to emphasizing the appearance of wave packet revivals, i.e., situations where a spreading wave packet reforms with close to its initial shape and width, we also examine in detail the approach to the collapsed phase where the position-space probability density is almost uniformly spread over the well. We focus on visualizing these phenomena in both position- and momentum-space as well as by following the time-dependent expectation values of and uncertainties in position and momentum. We discuss the time scales for wave packet collapse, using both an autocorrelation function analysis as well as focusing on expectation values, and find two relevant time scales which describe different aspects of the decay phase. In an Appendix, we briefly discuss wave packet revival and collapse in a more general, one-dimensional power-law potential given by V(k)(x)=V0|x/a|k which interpolates between the case of the harmonic oscillator (k=2) and the infinite well (k=∞).