Evolution of a continuously collapsed quantum system

Shizhong Mei

The standard quantum mechanics assumes Schrödinger equation for regular evolution and wave function collapse for measurement. As shown in this paper, only particular collapse equation can continuously transition to Schrödinge 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 Schrödinger evolution and collapse of equal duration.

Milan Dammjanović

O. Furtmaier

Le Hu, Andrew N. Jordan

Abstract Quantum mechanics admits two distinct evolutions: deterministic unitary dynamics governed by the Schrödinger equation and the probabilistic collapse of the wave function. We show that the continuous collapse of a quantum state under measurement can, on a trajectory-by-trajectory basis, be equivalently described as unitary evolution generated by a time- and state-dependent Hermitian Hamiltonian with stochastic parameters. While the ensemble dynamics remains non-unitary, each individual trajectory thus admits a unitary representation. We derive explicit forms of such Hamiltonians for projective measurements on arbitrary n -level systems and for continuous position measurements of a harmonic oscillator, and we propose experimental schemes to test these predictions. Our framework provides a new approach to modeling and controlling continuously monitored quantum systems using only state-dependent unitary resources.

Bei Zeng, Xie Chen, D. L. Zhou +1 more

Jincheng Zhang

Abstract: Quantum state collapse is a fundamental process in quantum mechanics, where a system transitions from a superposition of states to a definite state. This paper explores a novel recursive approach to this process, extending the concept of collapse to a continuous refinement of a system's state. We introduce a mechanism for calculating an "effective" collapse threshold based on the system's internal dynamics and dynamically adjusts the collapse rate to optimize the state's evolution. This design offers a departure from traditional static collapse theories, potentially leading to a more robust and dynamic understanding of quantum systems. The paper details the core mechanism, its mathematical underpinnings, and presents preliminary results demonstrating the effectiveness of this approach in a simplified quantum simulation.