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arXiv:physics/0208070

Quantum Phase Spase Representation for Double Well Potential

Dmytro Babyuk

Published Aug 19, 2002• 0 citations
arXivHTMLPDF

Abstract

A behavior of quantum states (superposition of two lowest eigenstates, Gaussian wave packet) in phase space is studied for one and two dimensional double well potential. Two dimensional potential is constructed from double well potential coupled linearly and quadratically to harmonic potential. Quantum trajectories are compared with classical ones. Preferable tunneling path in phase space is found. An influence of energy of initial Gaussian wave packet and trajectory initial condition on tunneling process is analyzed.. Coupling effect on trajectories and preferable tunneling path for two dimensional potential is studied.

DOI
10.48550/arxiv.physics/0208070

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A behavior of quantum states (superposition of two lowest eigenstates, Gaussian wave packet) in phase space is studied for one and two dimensional double well potential. Two dimensional potential is constructed from double well potential coupled linearly and quadratically to harmonic potential. Quantum trajectories are compared with classical ones. Preferable tunneling path in phase space is found. An influence of energy of initial Gaussian wave packet and trajectory initial condition on tunneling process is analyzed.. Coupling effect on trajectories and preferable tunneling path for two dimensional potential is studied.
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Abstract The Wigner-Moyal approach is applied to investigate the dynamics of the Gaussian wave packet moving in a double-well potential in the ‘Mexican hat’ form. Quantum trajectories in the phase space are computed for different kinetic energies of the initial wave packet in the Wigner form. The results are compared with the classical trajectories. Some additional information on the dynamics of the wave packet in the phase space is extracted from the analysis of the cross-correlation of the Wigner distribution function with itself at different points in time.
Abstract We perform quantum trajectory simulations of the decay dynamics of initially localized resonant states. Quantum dynamics is represented by a swarm of interacting trajectories which maps the originally quantum problem into the motion of an equivalent (higher-dimensional) classical system. We address two model problems, in which the decay of the initial resonance leads to either spatially confined or asymptotically free wave-packet dynamics, specifically on a double well potential and on a potential plain. The traditional choice of fixed boundary conditions in the interacting trajectory representation (ITR), set at infinity, is found to have a moderate influence on the accuracy of the ITR of quantum trajectory dynamics, for the motion on a double well potential, i.e. the results of the trajectory-based scheme are in good correspondence with those obtained via quantum wave-packet propagation up to several fundamental vibrational periods. On the other hand, standard boundary conditions have negligible effect on the interacting trajectory dynamics of a decaying shape resonance, whose predictions reproduce quantum mechanical results at long times.
We have studied wavepacket dynamics in the Razavy hyperbolic double-well (DW) potential which is coupled to a harmonic oscillator (HO) by linear and quadratic interactions. Taking into account the lowest two states of DW and (N+1)(N+1)(N+1) states of HO (N=1N=1N=1 to 10), we evaluate eigenvalues and eigenfunctions of the composite system. An analytical calculation is made for N=1N=1N=1 and numerical calculations are performed for 1<N≤101 < N \leq 101<N≤10. Quantum tunneling of wavepackets is realized between two bottoms of composite potential U(x,y)U(x,y)U(x,y) where xxx and yyy denote coordinates in DW and HO potentials, respectively. It has been shown that with increasing NNN and/or the coupling strength, the tunneling period is considerably increased. Phase space plots of ⟨x⟩\langle x \rangle⟨x⟩ vs. ⟨px⟩\langle p_x \rangle⟨px​⟩ and ⟨y⟩\langle y \rangle⟨y⟩ vs. ⟨py⟩\langle p_y \rangle⟨py​⟩ are elliptic, where ⟨⋅⟩\langle \cdot \rangle⟨⋅⟩ denotes an expectation value for the two-term wavepacket. This result is quite different from the relevant one previously obtained for the quartic DW potential with the use of the quantum phase space representation [Babyuk, arXiv:0208070]. Similarity and difference between results calculated for linear and quadratic couplings, and the uncertainty relation in the model are discussed.
The quantum dynamics of a particle in a double-well potential under a monochromatic external driving force is computed by using a minimum-uncertainty Gaussian wave packet as an initial state. Irregular or regular time development of dynamical variables is obtained, depending on whether the initial wave packet is located in the classical chaotic sea or the regular region in phase space, respectively. Tunnelings between nonresonant islands and between resonant islands are obtained. Tunneling times between nonresonant islands decrease smoothly, in general, with the increase of the amplitude of the driving force, but tunneling times between resonant islands are more erratic.