On Schrödinger-Hermite Operators in Lattice Quantum Mechanics

Hynek Bíla

Bernard H. Lavenda

A modification of Frenkel's time-dependent variational principle (TDVP) determines the complex phase of the wave function: the real and imaginary parts give the phase prescribed by the Schrodinger equation and the correct normalization, respectively. They are given by the hermitized and anti-hermitized components of the frequency operator. The quantum-mechanical TDVP is transcribed into a classical action principle where the hermitized and anti-hermitized components of the frequency operator are shown to be related to the velocity potential and stream function, respectively, of the complex potential method of two-dimensional hydrodynamic flow. The product of metric coefficients that appears in the Poisson brackets is the derivative of the circulation; the conservation of circulation, angular momentum, and particle number are all equivalent.

L. Piccirillo

In this chapter, the authors introduce the Hamilton-Jacobi equation showing that classical mechanics can be described with a wave equation for the classical action. In order to find the equation showing that classical mechanics already contains a wave behavior in matter particles, the people need to go back to the Lagrange/Hamilton formalism. Davisson-Germer experiment has shown without any doubt that electrons behave as massive particles whose dynamics is governed by some sort of wave. A free particle is apparently the conceptually simplest system to study because the TISE will reduce to its simplest form. The simple (linear) harmonic oscillator is of paramount importance in classical as well as quantum mechanics. A proper solution to the Harmonic oscillator problem can be found by solving Schrodinger equation analytically. Charles Hermite was a French mathematician who gave important contributions in many fields of mathematics used in physics like, for example, Hermitian operators and Hermite polynomials.

N. M. Bogoliubov, V. E. Korepin

F. G. Scholtz

Zhang Shu

The measurable mechanical-quantity operator of quantum mechanics has to be hermite operator.Every momentum operator which shows hermite property must obey any rules,while it is not hermite for all wave functions.It will play an important role in the studying and the understanding quantum mechanics for us if we can clarify the condition for the hermitianity of momentum operator.In this paper,the hermitianity of momentum operator had been well discussed term by term by some suitable examples.