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arXiv:2509.11832

Localised solution of stochastic Schrödinger equation describing simultaneous continuous measurement of position and momentum

Daniel Bedingham

Published Sep 15, 2025• 0 citations
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Abstract

A stochastic Schrödinger equation is presented to describe simultaneous continuous measurement of the position and momentum of a non-relativistic particle. The equation is solved to yield a state localised in position and momentum contingent on the uncertainty principle. The state is understood as that to which the particle tends after a period of continuous measurement. The solution takes the form of a wave packet of fixed Gaussian shape whose centre undergoes random motion in phase space. It is shown to be locally stable.

DOI
10.48550/arxiv.2509.11832

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A stochastic Schrödinger equation is presented to describe simultaneous continuous measurement of the position and momentum of a non-relativistic particle. The equation is solved to yield a state localised in position and momentum contingent on the uncertainty principle. The state is understood as that to which the particle tends after a period of continuous measurement. The solution takes the form of a wave packet of fixed Gaussian shape whose centre undergoes random motion in phase space. It is shown to be locally stable.
We derive the stochastic Schr\"odinger equation for the limit of continuous weak measurement where the observables monitored are canonical position and momentum. To this end we extend an argument due to Smolianov and Truman from the von Neumann model of indirect measurement of position to the Arthurs and Kelly model for simultaneous measurement of position and momentum. We require only unbiasedness of the detector states and an integrability condition sufficient to ensure a central limit effect. Despite taking a weak interaction as opposed to a weak measurement limit, the resulting stochastic wave equation is of the same form as that derived in a recent paper by Scott and Milburn for the specific case of joint Gaussian states.
The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.
We analyze the non-Markovian stochastic Schrödinger equation describing a particle subject to spontaneous collapses in space (in the language of collapse models), or subject to a continuous measurement of its position (in the language of continuous quantum measurement). For the first time, we give the explicit general solution for the free particle case (H=p(2)/2m) and discuss the main properties. We analyze the case of an exponential correlation function for the noise, giving a quantitative description of the dynamics and of its dependence on the correlation time.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
The question of phase measurement for a quantum oscillator using the quantum non-demolition principle (NDP), which does not need the use of the wave packet reduction postulate, is studied. It is shown that this approach is appropriate to introduce in a proper way the relevant observables for unlocalized (unsharp) measurement of quantum phase. Our analysis suggests that in contrast to other quantum-mechanical problems (e.g. The position and the momentum of a particle), the operator to represent the phase of a quantum field cannot be separated from the measurement process. Thus, the quantum phase operator must be defined for a corresponding measurement scheme. Moreover, it allows us to describe the time continuous process of quantum phase observations and also to study the stochastic dynamics of the oscillator under such measurement.