Quantum Trajectories based on the Weak Value

Abstract

The notion of trajectory of an individual particle is strictly inhibited in quantum mechanics because of the uncertainty principle. Nonetheless, the weak value, which has been proposed as a novel and measurable quantity definable to any quantum observable, can offer a possible description of trajectory on account of its statistical nature of the value. In this paper, we explore the physical significance provided by this weak trajectory by considering various situations where interference takes place simultaneously with the observation of particles, that is, in prototypical quantum situations for which no classical treatment is available. These include the double slit experiment and Lloyd's mirror, where in the former case it is argued that the real part of the weak trajectory describes an average over the possible classical trajectories involved in the process, and that the imaginary part is related to the variation of interference. It is shown that this average interpretation of the weak trajectory holds universally under the complex probability defined from the given transition process. These features remain essentially unaltered in the case of Lloyd's mirror where interference occurs with a single slit.

11 pages, 7 figures

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Cited by in corpus (9)

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• Identifying weak values with intrinsic dynamical properties in Modal theories
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• Advantages of the Kirkwood-Dirac distribution among general quasi-probabilities for finite-state quantum systems
• Control of particle propagation beyond the uncertainty limit by interference between position and momentum
• Weak values from path integrals
• Proposal to observe paths superpositions in a double-slit setup
• The solution to the Hardy's paradox
• Controlling and measuring a superposition of position and momentum