Time-extended measurement of the position of a driven harmonic oscillator

TL;DR

This paper analyzes how finite-duration measurements affect the observed position of a driven harmonic oscillator, revealing that the pointer indicates an average position influenced by external forces and measurement duration.

Contribution

It provides a detailed quantum mechanical analysis of time-extended position measurements for a driven harmonic oscillator, including the effects of external forces and measurement duration.

Findings

Pointer indicates average of initial and final positions with a force-dependent term.

Measurement results relate to transition probabilities of the undisturbed oscillator.

Approximate probability distribution matches that of the free oscillator at measurement end.

Abstract

The von Neumann interaction between a particle and an apparatus, both of arbitrary mass, has been considered in the measurement of the position of a simple harmonic oscillator acted on by an external force. When the measurement has finite duration, both the motion of the pointer and the oscillator influence the result of the measurement. Provided that the oscillator is in an eigenstate of its position at the start of the measurement, the pointer will indicate the arithmetic average between the initial and final position of the particle with an added term which depends on the duration of the measurement and the frequency of the oscillator. This additional term is determined by the external force which also causes the appearance of a phase factor in the wave function at the end of the measurement. This phase factor depends on the average of the initial and final positions of the particle. Furthermore, the probability that the pointer will indicate a given average value is equal to the transition probability for the undisturbed free oscillator to experience the change in position. If the initial state of the pointer is a narrow wavepacket, then for any initial state of the oscillator, the measurement yields, approximately, the undisturbed probability distribution for the position of the free oscillator at the end of the measurement. The transition probability for the pointer to experience a change in its position has also been evaluated.