Quantum Chaos for the Vibrating Rectangular Billiard

Abstract

We consider oscillations of the length and width in rectangular quantum billiards, a two "degree-of-vibration" configuration. We consider several superpositon states and discuss the effects of symmetry (in terms of the relative values of the quantum numbers of the superposed states) on the resulting evolution equations and derive necessary conditions for quantum chaos for both separable and inseparable potentials. We extend this analysis to n-dimensional rectangular parallelepipeds with two degrees-of-vibration. We produce several sets of Poincaré maps corresponding to different projections and potentials in the two-dimensional case. Several of these display chaotic behavior. We distinguish between four types of behavior in the present system corresponding to the separability of the potential and the symmetry of the superposition states. In particular, we contrast harmonic and anharmonic potentials. We note that vibrating rectangular quantum billiards may be used as a model for quantum-well nanostructures of the stated geometry, and we observe chaotic behavior without passing to the semiclassical (ℏ → 0) or high quantum-number limits.

Nonadiabatic dynamics in semiquantal physics

“…This is the approach that has been followed in the study of vibrating quantum billiards. 13,14,25 Equation (28) 13 We note that diabatic expansions are often more convenient for practical calculations because they correspond to fixed electronic states. Adiabatic expansions (and associated potential energy surfaces), on the other hand, arise naturally from quantum chemistry calculations and are also amenable to a dynamical systems approach.…”

“…In general, a molecular excited state can be decomposed into electronic, vibrational, and rotational excitations. Two examples of vibrational motion are pulsing (as in vibrating quantum billiards 14 ) and "bouncing" of the centerof-mass (as has been proposed as a mechanism for energy transfer in buckyballs 10 ). Together, the vibrational and rotational excitations comprise the nuclear (or rovibrational) contribution to the energy.…”

“…13 In semiquantal systems, many of which may be abstracted mathematically as vibrating quantum billiards, one often observes a form of quantum chaos known as semiquantum chaos, where the nomenclature reflects the fact that it occurs in the semiquantal regime. 14,25,27 Another form of quantum chaos, called quantized chaos or quantum chaology, is studied in the semiclassical and high quantum-number regimes. 1,28 This latter behavior may be observed by fully quantizing the motion of the molecular systems we have been describing.…”

“…10,11 These systems may be abstracted mathematically in terms of vibrating quantum billiards, 12,13 which are amenable to a semiquantal description because the boundaries are classical and the enclosed particles are quantum-mechanical. 14 We also survey other systems such as a two-level system interacting with the electromagnetic field of a laser cavity, 15 micromasers, 16 quantum particle-spin systems, 17 SQUIDs, 18 and nuclear collective motion. 19…”

“…More generally, a d-mode Galërkin expansion of a vibrating quantum billiard corresponds to a d-term quantum system coupled in a time-dependent, self-consistent fashion to r classical degrees-of-freedom.These classical dof correspond to the dov of the quantum billiard. Taking d = 2 and r = 1, one obtains a system precisely analogous to a diatomic molecule (such as NaCl) with two electronic states (of the same symmetry) coupled nonadiabatically by a single internuclear vibrational coordinate.Such symmetry, which can be expressed in terms of symmetry conditions in a superposition state's quantum numbers,13,14 are one possible cause of the electronic degeneracy discussed earlier. That is, degeneracy and neardegeneracy of electronic energy levels are often a consequence of a system's inherent symmetries.…”