Non-deterministic Quantum Mechanics, the Two-Slit Experiment, Measurement, Wave-Particle Duality, Spin, Mass and Energy1

Wave-Particle Duality, Spin, Mass and Energy1 Damon Wai Kwan So2 Oxford Centre for Mission Studies, Woodstock Road, Oxford OX2 6HR. Abstract A previous paper [1] proposed an alternative interpretation of quantum mechanics which is distinct from the Copenhagen Interpretation and the pilot wave theory. This alternative interpretation and its associated nondeterministic surfing velocity on the S (phase) surface are applied here to model point-like entities (particles and photons) with non-deterministic trajectories in the one-slit experiment and the two-slit experiment. The diffraction integral for these point-like entities, as a solution to the Helmholtz equation, is in the same form as the diffraction integral found in optical studies so that the mathematical results in optical studies regarding diffraction and interference patterns can be readily applied to the point-like entities modelled here. The nonrelativistic model for slowly moving particles and the relativistic model for fast moving particles and photons employed in this paper are therefore consistent with the the observed particle nature and the observed wavy behaviour. A crucial factor accounting for the common diffraction behaviour of the point-like entities (particles and photons) in the two models is that these models share the same Helmholtz equation (where the mass parameter does not appear) which gives rise to their identical diffraction integral. Therefore, these models resolve the problem of wave-particle duality for slowly moving particles, fast moving photons and fast moving particles. The disappearance of interference pattern when observation is made of the point-like entities in the two-slit experiment can be explained by an exchange of momentum between such an entity and the measuring agent (e.g., in scattering) such that the existing wave function goes through a sudden wholesale transformation, not collapse, in order to maintain consistency with the new momentum. The probability density distribution of the new wave function corresponding to the new momentum is much more localised; hence the disappearance of the interference pattern. Measurement is thus defined as an event where an exchange of momentum between the measured point-like entity and the measuring agent takes place. Hence, the two proposed models are consistent with the observed impact of measurement on the interference pattern. The two models also reproduce the observed de Broglie’s formula for momentum and wavelength, and the observed Planck-Einstein relation for energy and frequency. In addition, a non-deterministic component of the surfing velocity on the S (phase) surface in the models is set to be responsible for satisfying Born’s rule. As Born’s rule is satisfied, the angular momentum generated by the deterministic component of the surfing velocity is integrated over the relevant domain, yielding the expected spin value of the particle in consideration. Hence, the models reproduce the two aforementioned relations and the particle’s spin value. The probability density in Born’s rule is also interpreted as the non-dimensional time density or the time-averaged probability density such that the rule is satisfied over a given period of time, not at an instant. This implies that the quantum mechanical equations of the two models describe the time-averaged statistical behaviour of the particle, not its instantaneous motion; hence the room for non-determinacy in its instantaneous motion set within the bound of the time-averaged statistical constraint (Born’s rule). The notions of rest mass, rest energy and massless particles are investigated with reference to the particle’s surfing momentum on the S surface. A new definition of energy is proposed with partial support by data. If validated by experiments, this definition of energy could shed some light on the puzzle of dark matter. The crucial non-deterministic root in quantum mechanics is identified and Heisenberg’s Uncertainty Principle is critiqued. Finally, a possible relationship between a quantum-potential-like term and dark energy is posed. Mathematical Physics Preprint Archive, University of Texas,, June 2021