Quantum Mechanics Without a Hamiltonian Operator

Alessandro Sergi

The symplectic structure of Weinberg's formalism for nonlinear quantum mechanics is first unveiled and then generalized to introduce non-Hamiltonian quantum mechanics. By exploiting the correspondence between wave and matrix mechanics, a link between this generalization and a non-Hamiltonian commutator, proposed recently by this author, is found. The general correspondence between operator and wave formalisms in non-Hamiltonian quantum mechanics is exploited to introduce a quantum-classical theory of wave fields. This can be considered as a first step toward a deeper understanding of the relation between operator quantum-classical mechanics, introduced some time ago, and the original quantum-classical scheme of motion where wave functions are evolved in time and the classical degrees of freedom follows surface-hopping trajectories on single quantum states.

Lipo Wang, Robert F. O’Connell

Alexander N. Grigorenko

Juan M. Romero, Oswaldo Gonzalez-Gaxiola, J. Ruiz de Chávez +1 more

In this paper a quantum mechanics is built by means of a non-Hermitian momentum operator. We have shown that it is possible to construct two Hermitian and two non-Hermitian type of Hamiltonians using this momentum operator. We can construct a generalized supersymmetric quantum mechanics that has a dual based on these Hamiltonians. In addition, it is shown that the non-Hermitian Hamiltonians of this theory can be related to Hamiltonians that naturally arise in the so-called quantum finance.

D. Arsenović, Nikola Burić, Duska B Popovic +2 more

In the Hilbert space formulation of quantum mechanics, ideal measurements of physical variables are discussed using the spectral theory of Hermitian operators and the corresponding projector-valued measures (PVMs). However, more general types of measurements require the treatment in terms of positive-operator-valued measures (POVMs). In the Hamiltonian formulation of quantum mechanics, canonical coordinates are related to PVM. In this paper the results of an analysis of various aspects of applications of POVMs in the Hamiltonian formulation are reported. Several properties of state parameters and quantum observables given by POVMs or represented in an overcomplete basis, including the general Hamiltonian treatment of the Neumark extension, are presented. An analysis of the phase operator, given by the corresponding POVMs, in the Hilbert space and the Hamiltonian frameworks is also given.

David Z. Albert