Quantum Theory of the Harmonic Oscillator in Nonconservative Systems

P. Caldirola

Bowen Huang, 首都师范大学物理系，北京 100037

A time-dependent damped harmonic oscillator with a nonconservative force quadratic in velocity (THOFQV) is described by an appropriate Lagrangian,and hence the general solution of THOFQV can be found.The solution of the Schrdinger equation for the system,such as the eigenfunctions and eigenvalues,is derived by utilizing these invariant operators.

Carlo Jacoboni

Johan Noldus

We start by rewriting classical mechanics in a quantum mechanical fashion and point out that the only difference with quantum theory re- sides at one point. There is a classical analogon of the collapse of the wavefunction and an extension of the usual Born rule is proposed which might solve this problem. We work only with algebra’s over the real (com- plex) numbers, general number fields of finite characteristic allowing for finite dimensional representations of the commutation relations are not considered given that such fields are not well ordered and do not give rise to a well defined probability interpretation. Our theory generalizes however to discrete spacetimes and finite dimensional algebra’s. Looking at physics this way, spacetime itself distinguishes itself algebraically by means of well chosen commutation relations and there is further nothing special about it meaning it has also a particle interpretation just like any other dynamical variable. Likewise, there is no reason for the dynamics to be Hamiltonian and therefore we have a nonconservative formulation of quantum physics at hand. The harmonic oscillator (amongst few oth- ers) distinguishes itself because the algebra forms a finite dimensional Lie algebra; the classical and quantum (discrete) harmonic oscillator are stud- ied in a some more generality and examples are given which are neither classical, nor quantum.

Gerard McCaul, Alexander Nikolaevich Pechen, Denys I. Bondar

Applying the theory of self-adjoint extensions of Hermitian operators to Koopman von Neumann classical mechanics, the most general set of probability distributions is found for which entropy is conserved by Hamiltonian evolution. A new dynamical phase associated with such a construction is identified. By choosing distributions not belonging to this class, we produce explicit examples of both free particles and harmonic systems evolving in a bounded phase-space in such a way that entropy is nonconserved. While these nonconserving states are classically forbidden, they may be interpreted as states of a quantum system tunneling through a potential barrier boundary. In this case, the allowed boundary conditions are the only distinction between classical and quantum systems. We show that the boundary conditions for a tunneling quantum system become the criteria for entropy preservation in the classical limit. These findings highlight how boundary effects drastically change the nature of a system.