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arXiv:cond-mat/9611087

A new uncertainty Relation (revised version)

F. Ghaboussi

Published Nov 12, 1996• 0 citations
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Abstract

We formulate according to the quantum mechanical uncertainty relation a new quantum electrodynamical uncertainty relation $Δ\breve{A} . Δl \sim \hbar/e$ where $\breve{A}$ and $Δl \geq l_B$ are the electromagnetic pure gauge potential, the position uncertainty and the magnetic length, respectively. Then, we show that the observed potential drops on the edge of QHE samples are varifications of this uncertainty relation, where the quantum potential drop of the relevant component of potential can be considered as its quantum uncertainty $Δ\breve{A}$.

DOI
10.48550/arxiv.cond-mat/9611087

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We formulate according to the quantum mechanical uncertainty relation a new quantum electrodynamical uncertainty relation ΔA˘.Δl∼ℏ/eΔ\breve{A} . Δl \sim \hbar/eΔA˘.Δl∼ℏ/e where A˘\breve{A}A˘ and Δl≥lBΔl \geq l_BΔl≥lB​ are the electromagnetic pure gauge potential, the position uncertainty and the magnetic length, respectively. Then, we show that the observed potential drops on the edge of QHE samples are varifications of this uncertainty relation, where the quantum potential drop of the relevant component of potential can be considered as its quantum uncertainty ΔA˘Δ\breve{A}ΔA˘.
We discuss a model where the relation between Hall potential AHA_HAH​ = and gate potential AGA_GAG​, which is manifested by the stepped QHE = curve, is described by an uncertainty relation ΔAH.=ΔAG=3DB(ℏ/e)ΔA_H . = ΔA_G =3D B(\hbar/e)ΔAH​.=ΔAG​=3DB(ℏ/e).
Recently, S\'anchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}3\ensuremath{\hbar}/2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}$ which involves the mean value and the variance of the square of the angular momentum operator.
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Uncertainty relations are commonly praised as one of the central pillars of quantum theory. Usually, they are taught in the first weeks of a beginners lecture, and introduced in the ?rst chapters of a textbook. However, their precise operational meaning and a formulation in a general context, i.e. beyond the example of position and momentum observables, are often left out. The reasoning for this is twofold: On one hand, an exact operational definition of uncertainty, indeterminacy and a corresponding uncertainty principle has been the content of many debates since the early days of quantum mechanics until today. From a modern perspective, we have the consent that there are at least the two notions of preparation and measurement uncertainty: the first notion prohibits the existence of dissipation free states and the latter one the existence of error free joint measurements. On the other hand, we have that, the mathematical tools, which are needed for comprehensive treatment of uncertainty relations in a general context, are still under development and usually go far beyond the mathematical level of an introductory course. In this thesis we will investigate these two notions of uncertainty, their corre- sponding uncertainty relations, such as their interplay. The aim of this thesis is to give answers to the central questions: (1.) Which quantities should be used to formulate uncertainty? (2.) How can we compute uncertainty relations for those? (3.) Are there connections between the two notions of uncertainty? We will do this, whenever possible, in a most general context and with a focus on relevant examples, otherwise. Therefore, we will consider constructions of mea- surement errors and deviation measures that quantify uncertainty, based on, so called, cost functions. Commonly used uncertainty measures like variances, en- tropies, and the Hamming distance are examples for these. We will investigate the structure of the corresponding uncertainty relations and provide several methods that enable us to compute them. The third question is addressed by a theorem that shows, for sharp observables, that measurement uncertainty relations can be lower bounded by preparation uncertainty relations, whenever the same cost function is used.
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.