On the origin of preferred-basis and evolution pattern of wave function

TL;DR

This paper explores the connection between wave function collapse and Schrödinger evolution, deriving conditions under which collapse equations can transition to standard quantum mechanics and determining the preferred basis.

Contribution

It introduces a unique set of basis determination equations from collapse dynamics and shows how collapse equations can be conditioned to produce alternating Schrödinger evolution and collapse.

Findings

Collapse equations can continuously transition to Schrödinger equation.

Preferred basis depends on the system Hamiltonian and is uniquely determined.

Collapse must include a cyclic function for derivative continuity, leading to alternating evolution and collapse.

Abstract

The standard quantum mechanics assumes Schr\"odinger equation for regular evolution and wave function collapse for measurement. As shown in this paper, only particular collapse equation can continuously transition to Schr\"odinge equation. The collapse equation also adds some restriction to the preferred-basis. Under the assumptions that the preferred-basis depends on the whole system Hamiltonian but is not affected by the weights of the basis functions in the system wave function, a unique set of determination equations of the basis functions is derived from the collapse equation. The second order time derivative of the wave function is continuous at the end of the collapse. To make the derivative continuous at the beginning of the collapse, it is proved that the collapse equation has to contain a cyclic function with period twice the duration of the collapse, which leads to conditioned alternating Schr\"odinger evolution and collapse of equal duration.