Restrictions Imposed by the Wave Function on the Results of Particle Momentum Measurements

Abstract

Using the example of the quantum dynamics of a particle in a one-dimensional configuration space (OCS), it is shown that to know the wave function implies not only statistical restrictions on the measurement results: the integrand in the standard formula for calculating the average values of (self-adjoint) operators and the Schr\"{o}dinger equation for the modulus and phase of the wave function uniquely also define ' fields of operators' as functions of coordinate and time. A key role in establishing the physical meaning of these fields is played by the fact that the field of the kinetic energy operator contains two heterogeneous contributions: the first is determined by the field of the momentum operator, which is related only to the phase of the wave function, and the second coincides with the so-called "quantum mechanical potential", which is related only to the amplitude of the wave function. The values of these fields at each point of the OCS are considered as the average values of the corresponding observables for a pair of noninteracting particles (for a pair of systems of a one-particle ensemble). At each such point, the first contribution to the field of kinetic energy describes the kinetic energy of the center of mass of a pair of particles, and the second -- the energy of their motion relative to the center of mass. The field of the momentum operator and the field of the kinetic energy operator, taking into account the K\"{o}nig theorem, uniquely determine in the OCS two fields of particle momentum values at each point of the OCS. An analogue of the Heisenberg inequality for the deviations of both momentum fields from the field of the momentum operator is obtained.