The superposition principle and conservation laws in quantum mechanics

L.D. LANDAU, Efrat Lifshitz

Tian-Hai Zeng

The interaction between two parts in a compound quantum system may be reconsidered more completely than before and some new understandings and conclusions different from current quantum mechanics are obtained, including the conservation law in the evolution in an isolated quantum system, new understandings of duality of particle and wave and the superposition principle of states, three laws corresponding to Newton's laws, new understandings of measurement and the uncertainty relation, arguments against the non-locality of any entangled state and a simple criterion of coherence which is obtained for the experimenter to examine the correctness of the non-locality. These may make quantum mechanics be easily understood intuitively and some strange properties will not appear.

Jack Cohn

Hsiang Shun Chou

Abstract Physicists believe that there is a corresponding conservation law for each principle of symmetry. In standard textbooks on quantum mechanics, however, it is asserted that the time-reversal symmetry does not lead to any conservation law. There seems to be a missing piece of the jigsaw puzzle of the connection between the principles of symmetry and the conservation laws. In this paper, we resolve this difficulty by considering the time-reversal invariance of the transition probability a basic physical requirement of the time-reversal symmetry. We show that this requirement, together with the time-reversal covariance of the Schrödinger equation, guarantees the hermiticity of the Hamiltonian which ensures the probability conservation. Thus we complete the missing piece by demonstrating that the time-reversal symmetry leads to the probability conservation. The present paper can help undergraduate students gain deeper insight into the connection between the principles of symmetry and the conservation laws in quantum mechanics.

M. K. Bennett, David J. Foulis

Hans-Thomas Elze

Cellular automata (CA) can show well known features of quantum mechanics (QM), such as a linear updating rule that resembles a discretized form of the Schrödinger equation together with its conservation laws. Surprisingly, a whole class of “natural” Hamiltonian CA, which are based entirely on integer-valued variables and couplings and derived from an action principle, can be mapped reversibly to continuum models with the help of sampling theory. This results in “deformed” quantum mechanical models with a finite discreteness scale l, which for [Formula: see text] reproduce the familiar continuum limit. Presently, we show, in particular, how such automata can form “multipartite” systems consistently with the tensor product structures of non-relativistic many-body QM, while maintaining the linearity of dynamics. Consequently, the superposition principle is fully operative already on the level of these primordial discrete deterministic automata, including the essential quantum effects of interference and entanglement.