Fundamental Issues and Measurement Problem in Quantum Mechanics

The foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as \(c\)-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, the wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as the Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity that persists throughout an experiment and therefore does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions, like a Bose-Einstein condensate system, for which a single measurement yields a real ensemble. We comment on the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from relativistic quantum field theory.