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arXiv:1608.05774

Ambiguity appearing on the Hamiltonian formulation of Quantum Mechanics

Gustavo V. López, Ana Griselda, Carlos R. Martínez-Prieto

Published Aug 20, 2016• 1 citation
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Abstract

For the symmetric harmonic oscillator and the symmetric bouncer defined in 2-D, two different Hamiltonian are given describing the same classical dynamics; however, their quantum dynamics behavior are different.
DOI
10.48550/arxiv.1608.05774

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We study classical and quantum dynamics of the symmetric harmonic oscillator and the symmetric bouncer defined in 2-D. For these systems we get each of them for two different constants of motion, two Lagrangians and two Hamiltonians which describe the same classical dynamics. However, the quantization of these systems (using Schrödinger equation), using their two equivalents Hamiltonian, describes different quantum dynamics for each of them. This represents an ambiguity on the Hamiltonian formulation of the Quantum Mechanics.
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The dynamics of driven binary oscillators in the PT\mathcal{PT}PT-symmetric harmonic potential is investigated and the corresponding analytical solutions of the time-dependent wave function are obtained. Our numerical results exhibit that there is the population oscillation, which is different from the case of normal dissipation or gain systems. After studying the motion of driven harmonic oscillators which are coupled via the E⨂eE⨂eE⨂e Jahn-Teller effect in the Hermitian potential, we find that the dynamics behavior of these two systems is similar under certain conditions. As we know, the oscillation behavior is induced by the exchange interaction in the oscillator system with Jahn-Teller coupling. Our study shows that the binary quantum systems with PT\mathcal{PT}PT-symmetric Hamiltonians may be used to approximately describe the dynamics of coupling quantum systems and the quantum transport process in few-particle or other systems.