Differential conservation laws in nonrelativistic quantum mechanics

K. Kong Wan, D. R. E. Timson

K. S. Mallesh, S. Chaturvedi, V. Balakrishnan +2 more

L.D. LANDAU, Efrat Lifshitz

Bill Poirier, Hung-Ming Tsai

Abstract We present novel aspects of a trajectory-based theory of massive spin-zero relativistic quantum particles. In this approach, the quantum trajectory ensemble is the fundamental entity. It satisfies its own action principle, leading to a dynamical partial differential equation (via the Euler-Lagrange procedure), as well as to conservation laws (via Noether’s theorem). In this paper, we focus on the derivation of the latter. In addition to the usual expected energy and momentum conservation laws, there is also a third law that emerges, associated with the conditions needed to maintain global simultaneity. We also show that the nonrelativistic limits of these conservation laws match those of the earlier, nonrelativistic quantum trajectory theory [J. Chem. Phys. 136, 031102 (2012)].

S. Twareque Ali, James A. Brooke, Paul Busch +2 more

Working within the framework of stochastic quantum mechanics, we prove that a mild reality constraint on the radial part of the resolution generator renders the (probability) current, for a single particle with only positive (or only negative) energies and having arbitrary integer spin, conserved. The result is proven both nonrelativistically and relativistically. This extends some earlier results that had been obtained for spinless systems. We argue that the origin of such a conservation law is a consequence of the particular geometry of space–time predicated by the formalism of stochastic quantum mechanics.

Niels Henrik David Bohr