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International Journal of Modern Physics A

Position, momentum, and wave spreading in curved space

Mohammad Khorrami

Published Feb 27, 2021• 1 citation
Publisher

Abstract

The effect of the geometry (deviation from the flat space) on the quantum evolution of the momentum and position of a free particle is discussed. It is shown that beginning with a wave-packet of minimum uncertainty (a Gaussian wave), there is a usual increase in the product of the volume uncertainties in the momentum and position space, as seen in the quantum mechanics on a flat spaces. But there is also a contribution from geometry. The leading order of this contribution is calculated.

DOI
10.1142/s0217751x21500652

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The effect of the geometry (deviation from the flat space) on the quantum evolution of the momentum and position of a free particle is discussed. It is shown that beginning with a wave-packet of minimum uncertainty (a Gaussian wave), there is a usual increase in the product of the volume uncertainties in the momentum and position space, as seen in the quantum mechanics on a flat spaces. But there is also a contribution from geometry. The leading order of this contribution is calculated.
The spread of the wave-function, or quantum uncertainty, is a key notion in quantum mechanics. At leading order, it is characterized by the quadratic moments of the position and momentum operators. These evolve and fluctuate independently from the position and momentum expectation values. They are extra degrees of quantum mechanics compared to classical mechanics, and encode the shape of wave-packets. Following the logic that quantum mechanics must be lifted to quantum field theory, we discuss the quantization of the quantum uncertainty as an operator acting on wave-functions over field space and derive its discrete spectrum, inherited from the sl2\textrm{sl}_{2}sl2​ Lie algebra formed by the operators x^2\hat{x}^{2}x^2, p^2\hat{p}^{2}p^​2 and xp^\widehat{xp}xp​. We further show how this spectrum appears in the value of the coupling of the effective conformal potential driving the evolution of extended Gaussian wave-packets according to Schrödinger equation, with the quantum uncertainty playing the same role as an effective intrinsic angular momentum. We conclude with an open question: is it possible to see experimental signatures of the quantization of the quantum uncertainty in non-relativistic physics, which would signal the departure from quantum mechanics to a QFT regime?
We analyze the time-behaviour of the Gaussian wave packet of a quantum particle moving in the field of a linear force, undergoing a continuous nondemolition observation of its position. In particular, the time-dependence of position and momentum dispersions, as well as position-momentum uncertainty is investigated. The discussion includes the case of a free particle and initially coherent state of a harmonic oscillator. It is found that in the beginning of observation the momentum dispersion and position-momentum uncertainty always increase in contrast to the position dispersion. It is also proved that all considered functions asymptotically take the form of harmonic oscillations, decaying to finite values. For initially coherent states, the asymptotic bounds for all considered functions are given.
The paper deals with the motion of a Gaussian wave-packet in a parabolic potential field, contrasting the quantum-mechanical behaviour of the packet with that which follows from pure classical mechanics. The behaviour is entirely classical only if the initial phase of the space-wave-function is linear in the space coordinate; in which case the wave-packet is initially a ‘best-possible’ packet according to Heisenberg's uncertainty relationship. The paper concludes with a brief reference to the similar problem for a free particle. Note added in Proof, 16 February 1946. Through the courtesy of Mr M. S. Bartlett, we have been privileged to see the manuscripts of some work by Mr J. E. Moyal and himself, in which the motion of a probability wave-packet is discussed in more general terms than our own: and in which it appears that the motion of our wave-packet, which we have described as non-classical, is indeed actually classical provided that a certain correlation exists between momentum and position in the p, q-distribution function. With this knowledge we now find that classical and quantal predictions agree precisely provided that the initial classical probability function is given by this is a particular case of their more general results.
The properties of quantum mechanics with a discrete phase space are studied. The minimum uncertainty states are found and these states become the Gaussian wave packets in the continuum limit. With a suitably chosen Hamiltonian that gives free particle motion in the continuum limit, it is found that full or approximate periodic time evolution can result. This represents an example of revivals of wave packets that in the continuum limit is the familiar free particle motion on a line. Finally we examine the uncertainty principle for discrete phase space and obtain the correction terms to the continuum case.