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arXiv:quant-ph/0306080 · Journal of Physics A Mathematical and General

Uncertainty relations in curved spaces

Alexey V. Golovnev, Прохоров Лев Васильевич

Published Feb 4, 2004• 15 citations
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Abstract

Uncertainty relations for particle motion in curved spaces are discussed. The relations are shown to be topologically invariant. New coordinate system on a sphere appropriate to the problem is proposed. The case of a sphere is considered in details. The investigation can be of interest for string and brane theory, solid state physics (quantum wires) and quantum optics.

DOI
10.1088/0305-4470/37/7/017

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Uncertainty relations for particle motion in curved spaces are discussed. The relations are shown to be topologically invariant. New coordinate system on a sphere appropriate to the problem is proposed. The case of a sphere is considered in details. The investigation can be of interest for string and brane theory, solid state physics (quantum wires) and quantum optics.
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We consider the uncertainty relation between position and momentum of a particle on $ S^1 $ (a circle). Since $ S^1 $ is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on $ S^D $ (a DDD-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate quantum mechanics on a general manifold.
We consider the uncertainty relation between position and momentum of a particle on S1 (a circle). Since S1 is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on SD (a D-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate quantum mechanics on a general manifold.
Quantum uncertainty Spin I want to start this section with a description of particle spin. In addition to the usual properties of mass, charge, position and velocity, particles have a further parameter which describes their state of rotation. Physicists call this ‘spin’. We have already seen that in quantum physics it is impossible to measure simultaneously, in a precise way, both speed and position (recall the Heisenberg uncertainty relations). The same considerations apply to spin, which cannot be absolutely determined with arbitrary precision. The things that we can in principle find out are the rate of rotation (that is, the number of revolutions per second) and the component of this rotation in a given direction in space (for example, the angle of the axis of rotation relative to the direction of motion). It is important to understand that these two knowable quantities can take only certain quantised values. This is an aspect of quantum physics that we have already encountered when describing wave motion. We can use as an analogy the stable vibrations that can be sustained on a piano string: the basic note is generated when the centre of the string vibrates to its maximum extent while the two ends are (necessarily) fixed.
A brief review of various existing mathematical formulations of the uncertainty relations in quantum mechanics, containing variances of two or more non-commuting operators, is given. In particular, inequalities for the products of higher-order moments of a coordinate and a momentum are considered, as well as inequalities making the uncertainty relations more accurate when additional information about a quantum system is available (for example, the correlation coefficient or the degree of mixing of a quantum state characterized by the trace of the squared statistical operator). The special cases of two, three, and four operators are discussed in detail.
For description of the quantum dynamics on a curved group manifold the path integrals in a space of the group parameters is offered. The formalism is illustrated by the HHH-atom problem.