Collapse Time for the Bohm-Bub Hidden Variable Theory

Jens Peter Christensen, R. D. Mattuck

R. D. Mattuck

R. D. Mattuck

Selçuk Ş. Bayın

J. H. Tutsch

A recently proposed theory of measurement for nonrelativistic quantum mechanics, due to Bohm and Bub, is generalized, and a mathematical analysis of the theory is presented. The fact that there are always theories of the Bohm-Bub type, given the standard formulation of quantum mechanics, is discussed. The phenomenological coefficient $\ensuremath{\gamma}$ of the original theory is defined to be the absolute value of the energy change of the system during measurement divided by $\ensuremath{\hbar}$. This definition leads to a Heisenberg-like relation between the average energy change of the system during the measurement process and the collapse time, but the new relation depends on the so-called hidden variables, and hence could lead to non-quantum-mechanical effects. The time dependence of the hidden variables is considered, and they are then assumed to be constant for each individual system. An experiment is proposed which would test this assumption. A positive result would disagree significantly with the quantum-mechanical prediction. The question of whether or not the new variables must remain unobservable is analyzed mathematically. A rigorous definition of simultaneous measurement is made, and the resulting interference is given a mathematical description.

J. H. Tutsch

The mathematical structure of the simplest nontrivial hidden variable theory is derived. The core of the resulting theory is an essentially unique nonlinear differential equation which gives a causal description of the continuous collapse of the wavefunction during a measurement. It is shown that the simplest collapse equation is the same as the equation given ad hoc by Bohm and Bub.