Derivation of Born's Rule as a Limiting Case of the Relative Frequency of Detection Using Single-Quamtum Events

Abstract

A new 'wave-particle non-dualistic interpretation of quantum mechanics at a single-quantum level' is presented by interpreting the Schrodinger wave function as an 'instantaneous resonant spatial mode' (IRSM) to which a quantum is confined and moves akin to the case of a test particle in the curved space-time of the general theory of relativity. This union of the wave and particle natures into a single entity is termed as non-duality. Using quantum formalism, the IRSM is shown to induce dual-vectors at the boundaries and interacts according to the inner-product. The overall phase associated with the state vector, which never contributes to the inner-product, is related to a particular eigenstate of an observable where the particle resides. This eigenstate becomes the natural outcome during observation of a single-quantum's single event. Observation over a large number of identical quanta, differing only by overall phases, results in the relative frequency of detection which yields Born's rule as a limiting case proving the absence of measurement problem in quantum mechanics. The non-duality not only provides the actual mechanism for the 'wave function collapse' but also statistically becomes equivalent to the Copenhagen interpretation. An explicit derivation for the Born rule using individual quantum events is provided by considering an example of spin-1/2 particles in the Stern-Gerlach experiment. In this regard, a generalized representation for the SU (2) algebra, facilitating the description of single-quantum events, is constructed without any deviations from the quantum formalism.