Solutions of the equations of motion in classical and quantum theories

Theoretical Physics

Konstantin Alekseevich Sveshnikov

Eugene P. Gross

We study the interaction of a nonrelativistic particle with a scalar field, with particular application to the theory of polarons. The approach is based on a general classical method for the integration of equations of motion. The Hamiltonian is transformed by successive canonical transformations, the first corresponding to describing the motion relative to special solutions of the equations of motion. This stage as applied to suitably ordered Heisenberg equations of motion is identical with intermediate coupling theory. The second transformation treats the coupled small oscillations of particle and field oscillators about the chosen special solution. This affords a natural extension of intermediate coupling theory for this problem. Differences between the classical and quantum theories arise in the ordering of operators; the differences play a crucial role in determining the effective cutoff in wave vector space.

Albert Huber, Paul Schreivogl

This work investigates the dynamics of closed quantum systems in the Bloch vector representation using methods from rigid body dynamics and the theory of integrable systems. To this end, equations of motion for Bloch components are derived from the von Neumann equation that are mathematically equivalent to equations of motion for a distribution of point masses from classical mechanics. Furthermore, using the Heisenberg equation, another system of Bloch vector equations is derived, which constitutes an Euler-Poinsot system of the type commonly encountered in the theory of torque-free spinning tops. This insight is used to prove the Liouville integrability of the corresponding Hamilton equations of motion and to construct an explicit closed-form solution using the Arnold-Liouville theorem. Within the same framework, stability criteria for quantum dynamics are then derived which correspond to the Energy-Casimir method of classical Newtonian mechanics. Following that, specific solutions to the equations of motion are constructed that encode the complex dynamics of composite quantum systems. Eventually, to show that this formalism provides concrete physical predictions, an analogue of the intermediate axis theorem is derived and the effect of oscillating entanglement is discussed. As a basis for this, special types of solutions to the equations of motion are derived that constitute oscillating entangled states, i.e., dynamical quantum states that change their entanglement structure from maximally entangled to separable and vice versa.

Д. Я. Петрина, V. I. Gerasimenko, V. Z. Enolski

J.-L. Gervais