Principles of quantum mechanics : nonrelativistic wave mechanics with illustrative applications

ANTON Z. CAPRI

Peter Sigmund, George Basbas

A variational functional has been derived for the expectation value of the energy of a quantum system governed by a time-dependent Hamiltonian. Application to a stationary system perturbed by a time-dependent force shows that a perturbational wave function correct to first order in the perturbation, when inserted into the functional, delivers an expression for the energy that is correct up to third order in the perturbation. The use of the variational functional by means of a trial function is illustrated on the forced harmonic oscillator where well-known results are reproduced easily.

Walter T. Grandy

S. Rajasekar, R. Velusamy

Most of the applications of quantum mechanics are in the realm of the three phases of matters: gases, liquids and solids. In these cases a nonrelativistic quantum theory is sufficient. However, in principle, a strictly correct treatment should include relativity. For relativistic particles, that is, for particles travelling at speeds close to the speed of light, the nonrelativistic Schrödinger equation is not applicable. Oskar Klein and Walter Gordon1 proposed a relativistic wave equation for a free particle. Their equation fell into dispute because the probability density function is not positive definite, thereby preventing the probability interpretation.

ANTON Z. CAPRI

Yuri A. Rylov

Quantum systems are dynamic systems restricted by the principles of quantum mechanics (linearity of dynamic equations, linear transformation of the wave function etc.). One suggests to investigate the quantum systems simply as dynamic systems, ignoring the quantum principles and constraints imposed by them. Such dynamic methods of investigation appear to be more adequate and effective, than the conventional quantum methods of investigation. Using these methods, which ignore the quantum principles, one can overcome the principal problem of quantum field theory: join of nonrelativistic quantum principles with the relativity principles. Investigation of dynamic system, described by the Schroedinger equation, admits one to prove that the Copenhagen interpretation is incompatible with quantum mechanics formalism. Besides, it is shown that sometimes the application of quantum principles leads to incorrect results.