Hamiltonian aspects of quantum theory

Päivi Törmä, Stig Stenholm, Igor Jex

We discuss the theory of extracting an interaction Hamiltonian from a preassigned unitary transformation of quantum states. Such a procedure is of significance in quantum computations and other optical information processing tasks. We particularize the problem to the construction of totally symmetric 2N ports as introduced by Zeilinger and his collaborators [A. Zeilinger, M. Zukowski, M. A. Horne, H. J. Bernstein, and D. M. Greenberger, in Fundamental Aspects of Quantum Theory, edited by J. Anandan and J. J. Safko (World Scientific, Singapore, 1994)]. These are realized by the discrete Fourier transform, which simplifies the construction of the Hamiltonian by known methods of linear algebra. The Hamiltonians found are discussed and alternative realizations of the Zeilinger class transformations are presented. We briefly discuss the applicability of the method to more general devices.

Dominic Branford, Oscar Dahlsten, Andrew J. P. Garner

Randal L. Hallford, Preet Sharma

The Non-Hermitian aspect of Quantum Mechanics has been of great interest recently. There have been numerous studies on non-Hermitian Hamiltonians written for natural processes. Some studies have even expressed the hydrogen atom in a non-Hermitian basis. In this paper the principles of non-Hermitian quantum mechanics is applied to both the time independent perturbation theory and to the time dependant theory to calculate the Stark effect. The principles of spherical harmonics has also been used to describe the development in the non-Hermitian case. Finally, the non-Hermitian aspect has been introduced to the well known Stark effect in quantum mechanics to find a condition in which the Stark effect will still be true even if a non-Hermitian Hamiltonian is used. This study completes the understanding at a fundamental level to understand the well known Stark effect. Doi: 10.28991/esj-2020-01242 Full Text: PDF

Domenico Giulini

Inge Svein Helland

Tudor Stefan Ratiu, Oleg Georgievich Smolyanov

This paper discusses the Hamiltonian aspects of the quantum theory of constrained Hamiltonian systems; in particular, we extend results of [9, 6] to such systems. In this paper, we usually refer to these systems as Hamilton–Dirac systems, because it was Dirac who first defined them (under the name of generalized Hamiltonian systems). Such systems arise in the application of the Legendre transform to unconstrained Lagrangian systems with singular Lagrange function (for which the Legendre transform is noninvertible). It should be emphasized that constrained Hamiltonian systems bear no relation to Lagrangian systems with holonomic constraints; however, them can be regarded as objects dual to Lagrangian systems with nonholonomic constraints. Namely, the Legendre transform turns Lagrangian systems with nonholonomic constraints into unconstrained Hamiltonian systems with singular Hamilton function