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Nanosystems Physics Chemistry Mathematics

Kicked particle dynamics in quantum graphs

Jambul Yusupov, Valijon Eshniyozov, O. Karpova, Dilmurod Saidov

Published Dec 21, 2015• 1 citation
Publisher

Abstract

The quantum dynamics of a delta-kicked driven particle in a star-shaped network is studied by obtaining an exact solution for the time-dependent Schrödinger equation within a single kicking period.The timedependence of the average kinetic energy and the Gaussian wave packet evolution are analyzed.
DOI
10.17586/2220-8054-2015-6-6-773-778

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Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
Using Gaussian wave packet solutions, we examine how the kinetic energy is distributed in time-dependent 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. We find, for specific choices of initial parameters, that as much as 90% of the kinetic energy can be localized (at least conceptually) in the `front half' of such Gaussian wave packets, and we visualize these effects.