Non-Relativistic Quantum Theory

David K. Nartonis

Boon Leong Lan, Mehdi Pourzand, Rui Jian Chu

To study quantum dynamics in the non-relativistic regime, the standard practice is to use non-relativistic quantum mechanics, instead of the relativistic theory, because it is thought the approximate non-relativistic result is always close to the relativistic one. Here we present a theoretical argument that this expectation is not true in general. In addition, supporting numerical evidence for the free rotor and hydrogen atom also shows the agreement between the two theories can break down quickly. For the radial Rydberg wave packet in hydrogen atom, the breakdown can occur before spontaneous emission and thus could be tested experimentally. Our surprising result shows relativistic quantum mechanics must be used, instead of the approximate non-relativistic theory, to correctly study quantum dynamics in the non-relativistic regime after the breakdown time. This paradigm shift opens a new avenue of research in a wide range of fields from atomic to molecular, chemical and condensed-matter physics.

Richard M. Sillitto, Robert C. Whitten

R.E. Moss

Luis Grave de Peralta

Background : The theory of a Fermi gas constitutes a useful physical model with multiple applications from solid state to stellar physics. The theory of a non-relativistic Fermi gas is relatively simple when compared with the one for a relativistic Fermi gas. Therefore, it would be useful if a rigorous theory of a relativistic Fermi gas could be constructed like the theory of a non-relativistic Fermi gas. Methods : Such a theory, for T > 0 K, is presented here. It is based on using a recently reported formula for the energies of a relativistic spin-0 particle in a three-dimensional infinite well. The Pauli exclusion principle is used for taking care of the fermion nature of the particles forming a Fermi gas. Results : A unified description of a Fermi gas at low temperatures is presented. This description is valid from the non-relativistic to the ultra-relativistic limits. Moreover, it is simple. The theory follows the relatively simple approach often used for constructing the theory of a non-relativistic Fermi gas. Conclusions : The ultra-relativistic results reported here are often obtained using a different approach than the one used in this work. These results are not new, but they corroborate the correctness of the presented approach. In addition, the relative simplicity of the approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses.

Ali Shojai, Fatimah Shojai

The standard relativistic de-Broglie–Bohm theory has the problems of tacyonic solutions and the incorrect non-relativistic limit. In this paper we obtain a relativistic theory, not decomposing the relativistic wave equations but looking for a generalization of non-relativistic Bohmian theory in such a way that the correct non-relativistic limit emerges. In this way we are able to construct a relativistic de-Broglie–Bohm theory both for a single particle and for a many-particle system. At the end, the theory is extended to the curved space-time and the connection with quantum gravity is discussed.