Quantisation of the damped free particle

Eqab M. Rabei, Ahmed Al-Jamel

Arthur Charlesby

Shingo Ohtsuka, Susumu Kurihara

Craig M. Savage, Daniel F. Walls

We use a Markovian master equation to solve for the dynamics of a free particle damped by the coupling of its coordinate to a bath of harmonic oscillators. In the coordinate basis the reduced density matrix for the free particle tends to be diagonalized. Considering the evolution of an initial superposition of two plane waves, we find that the interference-pattern fringe contrast is decreased by the damping. However, for fixed times and sufficiently large damping, the fringe contrast increases with increasing damping. Considering an initial coordinate-space Gaussian wave function, we find that for sufficiently large damping the coupling to the environment models a quantum measurement of the particle's coordinate.

Pierre VAN BAAL

As position is no longer a quantum observable but free particles do not seem to be in contradiction with relativistic invariance, we can try to introduce such a free particle as a quantum observable. This observable is hence described by a plane wave ( 2.1 ) φ k → ( x → , t ) = e − i ( k → o t − x → ⋅ k → ) / ℏ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429073601/5d786806-46dc-40d1-b4d3-a3e5f58ba76e/content/equ_31.tif"/>

J. J. Powis

We give a brief review of geometric quantisation up to and including the Blattner-Kostant-Sternberg kernal. In general this leads to symmetric operators that are not essentially self-adjoint so motivating a study of Hermitian operators as observables in a generalised quantum mechanics. We show that a generalised squaring axiom can reproduce the results of Blattner-Kostant-Sternberg quantisation. We also show that quantisation with respect to polarisations with compact leaves gives results that conflict with the nonlocal nature of quantum mechanics. We develop a front form quantum mechanics of a free scalar particle using geometric quantisation. The front and instant forms are related via unitary maps derived from the pairing which intertwines quantisations with respect to the forms. The front form position operator has a maximally symmetric component so we are compelled to work within the framework of a generalised quantum mechanics; the result in there being no Hegerfeldt type instantaneous spreading of initially localised wavefunctions in the front form. Finally we show that this model can be lifted to a many particle free field theory.