Describing the Wave Function Collapse Process with a State-dependent Hamiltonian

Abstract: It is well-known that quantum mechanics admits two distinct evolutions: the unitary evolution, which is deterministic and well described by the Schr\"{o}dinger equation, and the collapse of the wave function, which is probablistic, generally non-unitary, and cannot be described by the Schr\"{o}dinger equation. In this paper, starting with pure states, we show how the continuous collapse of the wave function can be described by the Schr\"{o}dinger equation with a stochastic, time-dependent Hamiltonian. We analytically solve for the Hamiltonian responsible for projective measurements on an arbitrary $n$-level system and the position measurement on an harmonic oscillator in the ground state, and propose several experimental schemes to verify and utilize the conclusions. A critical feature is that the Hamiltonian must be state-dependent. We then discuss how the above formalism can also be applied to describe the collapse of the wave function of mixed quantum states. The formalism we proposed may unify the two distinct evolutions in quantum mechanics.