Quantum Solutions in Relativistic Classical Mechanics

F. Reuse

В. В. Ласуков, Т. В. Ласукова, M. O. Abdrashitova

Quantum solutions of the classical equation of relativistic mechanics are found. The synthesis of classical and quantum physics can become the basic formalism for the second quantum revolution, since the existence of quantum solutions to all equations of classical physics means that macroscopic bodies of both inanimate and living matter, under certain conditions, can be quantum objects. This new direction of physics can find application in the development of nature-like technologies.

Frank Blume

We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conservation equations for (probability) density functions give rise to velocity-field divergence equations the solutions of which generate-by way of superposition-the totality of solutions of various well-known classical and quantum-mechanical wave equations.

Lawrence Paul Horwitz

Kenneth G. Dyall, Knut Fægri

The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.

Mark Goodall