The main idea of "Quantum Chaos" studies is that Quantum Mechanics introduces two energy scales into the study of chaotic systems: One is obviously the mean level spacing $Δ\propto\hbar^d$ , where $d$ is the dimensionality; The other is $Δ_b\propto\hbar$ , which is known as the non-universal energy scale, or as the bandwidth, or as the Thouless energy. Associated with these two energy scales are two special quantum-mechanical (QM) regimes in the theory of driven system. These are the QM adiabatic regime, and the QM non-perturbative regime respectively. Otherwise Fermi golden rule applies, and linear response theory can be trusted. Demonstrations of this general idea, that had been published in 1999, have appeared in studies of wavepacket dynamics, survival probability, dissipation, quantum irreversibility, fidelity and dephasing.The main idea of "Quantum Chaos" studies is that Quantum Mechanics introduces two energy scales into the study of chaotic systems: One is obviously the mean level spacing $Δ\propto\hbar^d$, where $d$ is the dimensionality; The other is $Δ_b\propto\hbar$, which is known as the non-universal energy scale, or as the bandwidth, or as the Thouless energy. Associated with these two energy scales are two special quantum-mechanical (QM) regimes in the theory of driven system. These are the QM adiabatic regime, and the QM non-perturbative regime respectively. Otherwise Fermi golden rule applies, and linear response theory can be trusted. Demonstrations of this general idea, that had been published in 1999, have appeared in studies of wavepacket dynamics, survival probability, dissipation, quantum irreversibility, fidelity and dephasing.

arXiv:quant-ph/0201088Jan 20, 20020 citations

Energy Non-conservation in Quantum Mechanics

Sean M. Carroll, Jackie Lodman

arXiv:2101.11052Foundations of PhysicsJul 31, 202121 citations

Elements of Quantum Mechanics

WORLD SCIENTIFIC eBooksJun 1, 20106 citations

Supersymmetric quantum mechanics and statistical physics: Schrödinger-like thermodynamic equation

J. Morales, José Juan Peña

Supersymmetric quantum mechanics (SUSY QM) is used to determine the class of potentials that are solvable as well as to find their isospectral partners. To do that SUSY QM assumes the knowledge of the Witten superpotential as a particular solution of the Riccati equation involved in order to obtain the corresponding generalizations. Consequently, due to the fact that in statistical mechanics the internal energy is indeed known as a single temperature derivative of the Helmholtz free energy, a simple transformation has been proposed here in order to get, in an heuristic way, a Schrödinger-like thermodynamic (SLT) equation. In this proposal, the quantum Witten superpotential and the wavefunction correspond, in statistical physics, to the internal energy and to the canonical partition function respectively. With that, it is shown how all the results from SUSY QM can be incorporated in the statistical case, to find the generalized internal energy and partition functions as well as to generalize all the other related thermodynamical properties. As an example of the usefulness of the proposed SLT equation, the case of Boltzmann statistics has been considered explicitly and it was found that the corresponding generalized internal energy contain, as a particular case, the Bose–Einstein or the Fermi–Dirac probability distributions.Supersymmetric quantum mechanics (SUSY QM) is used to determine the class of potentials that are solvable as well as to find their isospectral partners. To do that SUSY QM assumes the knowledge of the Witten superpotential as a particular solution of the Riccati equation involved in order to obtain the corresponding generalizations. Consequently, due to the fact that in statistical mechanics the internal energy is indeed known as a single temperature derivative of the Helmholtz free energy, a simple transformation has been proposed here in order to get, in an heuristic way, a Schrödinger-like thermodynamic (SLT) equation. In this proposal, the quantum Witten superpotential and the wavefunction correspond, in statistical physics, to the internal energy and to the canonical partition function respectively. With that, it is shown how all the results from SUSY QM can be incorporated in the statistical case, to find the generalized internal energy and partition functions as well as to generalize all the other related thermodynamical properties. As an example of the usefulness of the proposed SLT equation, the case of Boltzmann statistics has been considered explicitly and it was found that the corresponding generalized internal energy contain, as a particular case, the Bose–Einstein or the Fermi–Dirac probability distributions.

Physica ScriptaJun 22, 20065 citations

A Generalized Analytical Mechanics in which Quantum Phenomena Appear

Yasuda, Masao

Currently, dynamics of a massive macroparticle is given by classical analytical mechanics (CM), while that of a massive micro one is given by quantum mechanics (QM). We propose a mechanics effective for both: We transform, under coordinate transformation, the covariant tensor of order two underlying the kinetic energy term of the Hamilton-Jacobi (H-J) eq. of CM, not with the action of the diffeomorphism group, but with an action of an extended diffeomorphism group. Then, the H-J eq., a first-order partial differential eq., is modified to a third-order one. The Euler-Lagrange eq. of CM, a second-order ordinary differential eq., related to the H-J eq. through action integral of the least action principle is accordingly modified to a fourth-order one. We thus obtain a mechanics which, because of the higher-order eqs., accommodates phenomena corresponding to quantum phenomena. Energy of a particle in a confining potential is quantized to the same values as those of QM. Particle distribution in an ensemble disagrees with that of QM, but agrees with experiments within experimental uncertainty. The mechanics therefore is a testable alternative to QM.Currently, dynamics of a massive macroparticle is given by classical analytical mechanics (CM), while that of a massive micro one is given by quantum mechanics (QM). We propose a mechanics effective for both: We transform, under coordinate transformation, the covariant tensor of order two underlying the kinetic energy term of the Hamilton-Jacobi (H-J) eq. of CM, not with the action of the diffeomorphism group, but with an action of an extended diffeomorphism group. Then, the H-J eq., a first-order partial differential eq., is modified to a third-order one. The Euler-Lagrange eq. of CM, a second-order ordinary differential eq., related to the H-J eq. through action integral of the least action principle is accordingly modified to a fourth-order one. We thus obtain a mechanics which, because of the higher-order eqs., accommodates phenomena corresponding to quantum phenomena. Energy of a particle in a confining potential is quantized to the same values as those of QM. Particle distribution in an ensemble disagrees with that of QM, but agrees with experiments within experimental uncertainty. The mechanics therefore is a testable alternative to QM.

arXiv:1003.0743Mar 3, 20100 citations

Elements of Quantum Mechanics

Ruslan P. Ozerov, Anatoli A. Vorobyev

Elsevier eBooksJan 1, 20078 citations

The main idea of "Quantum Chaos" studies is that Quantum Mechanics introduces two energy scales into the study of chaotic systems: One is obviously the mean level spacing

Δ\propto\hbar^d

, where

d

is the dimensionality; The other is

Δ_b\propto\hbar

, which is known as the non-universal energy scale, or as the bandwidth, or as the Thouless energy. Associated with these two energy scales are two special quantum-mechanical (QM) regimes in the theory of driven system. These are the QM adiabatic regime, and the QM non-perturbative regime respectively. Otherwise Fermi golden rule applies, and linear response theory can be trusted. Demonstrations of this general idea, that had been published in 1999, have appeared in studies of wavepacket dynamics, survival probability, dissipation, quantum irreversibility, fidelity and dephasing.