Position and Momentum in Non-Relativistic Quantum Mechanics

Pankaj Sharan

Pankaj Sharan

Ed Seidewitz

This note is an addendum to quant-ph/0507115. In that paper, I present a formalism for relativistic quantum mechanics in which the spacetime paths of particles are considered fundamental, reproducing the standard results of the traditional formulation of relativistic quantum mechanics and quantum field theory. Now, it is well known that there are issues with the ability to localize the position of particles in the usual formulation of relativistic quantum mechanics. The present note shows how, in the spacetime path formalism, the natural representation of on-shell 3-momentum states is effectively a Foldy-Wouthuysen transformation of the traditional representation, addressing the localization issues of position states and, further, providing a straightforward non-relativistic limit.

Chee Leong Ching, Rajesh R. Parwani

We construct the exact position representation for a deformed (non-relativistic) quantum mechanics which exhibits an intrinsic maximum momentum and use it to study problems such as a particle in a box and an asymmetric well. In particular, we show that unlike usual quantum mechanics, the present deformed case delays the formation of bound states in a finite potential well, a distinguishing feature that might be relevant for empirical investigations. We also contrast our results with the string-motivated type of deformed quantum mechanics which incorporates a minimum position uncertainty rather than a maximum momentum.

W. M. Stuckey, Michael Silberstein, Michael Cifone

We introduce a new interpretation of non‐relativistic quantum mechanics (QM) called Relational Blockworld (RBW). We motivate the interpretation by outlining two results due to Kaiser, Bohr, Ulfeck, Mottelson, and Anandan, independently. First, the canonical commutation relations for position and momentum can be obtained from boost and translation operators, respectively, in a spacetime where the relativity of simultaneity holds. Second, the QM density operator can be obtained from the spacetime symmetry group of the experimental configuration exclusively. We show how QM, obtained from relativistic quantum field theory per RBW, explains the twin‐slit experiment and conclude by resolving the standard conceptual problems of QM, i.e., the measurement problem, entanglement and non‐locality.

J. S. Faulkner