On the Energy Commutators in Quantum Mechanics

Neal H. McCoy

On commutation formulas in the algebra of quantum mechanics

Joseph V. Lepore

Vinod Kumar Khanna

An overview of main operators in quantum mechanics is presented. Operators for position, linear momentum, kinetic energy, potential energy, total energy, and angular momentum are introduced. Properties of linear operators are described. Commutative property of operators has profound implications because a pair of non-commutative operators cannot be measured accurately at the same time. Eigenfunctions and eigenvalues of an operator are defined, and operator form of Schrodinger’s equation is given. Most operators in quantum mechanics are of the Hermitian category because they always have real eigenvalues. The condition to test Hermitian nature of operators is specified. Postulates of quantum mechanics are stated to serve as a convenient starting point.

Abdelmadjid Maireche

Modified Schrodinger equation plays a fundamental role for describing the behavior of a particle at microscopic scale. In this context, we have studied the non-relativistic energy spectra of the modified quantum mechanical gravitational potential plus the harmonic oscillator (MQMGHO) potential using the generalized Bopp’s shift method and standard perturbation theory within the framework of non-commutative 3-dimensional phase space (NC: 3D-RSP) symmetries. Furthermore, we have shown that the modified Hamiltonian operator containing a perturbed Coulomb potential, which physically means the global Hamiltonian of MQMGHO potential, can be described as Hydrogenic atoms interacting with strong potential composed of quantum mechanical gravitational potential plus the harmonic oscillator (QMGHO) potential and an auxiliary part. The bound state energy eigenvalues, in terms of discreet atomic quantum numbers ( j , n, landm ), four infinitesimal parameters , , ,  induced by positionposition and phase-phase non-commutativity, in addition to the dimensional parameters of MQMGHO potential and the corresponding noncommutative Hamiltonian operator were obtained for Hydrogenic atoms. Moreover, we have shown that, the total complete degeneracy of energy

Yoshiaki Hashimoto

Esteban Marulanda, Alejandro Restrepo, J. Restrepo

In classical physics, there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics, due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, the energy is not equally distributed. We propose a correspondence between what is known as the classical energy equipartition theorem and its counterpart in the phase-space formulation in quantum mechanics based on the Wigner representation. Further, we show that in the high-temperature regime, the classical result is recovered.