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Chapter 12 : Heisenberg's uncertainty relation

史郎 石川

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

It may be usually considered that the (12.2) is the mathematical representation of the (12.3).However, it is not true.In fact, in [84], J. von Neumann pointed out the difference between Robertson's uncertainty relation (= (12.2)) and Heisenberg's uncertainty relation (= (12.3)).

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Chapter 12 : Heisenberg's uncertainty relation

シロウ イシカワ, Shiro Ishikawa, 史郎 石川

12.1 Introduction 12.2 Example due to Arthurs-Kelley 12.3 Approximate simultaneous measurement 12.4 Lemmas 12.5 Existence theorem 12.6 Uncertainty relations 12.7 EPR-experiment and Heisenberg's uncertainty relation

•Jan 1, 2006•0 citations
Abstract The uncertainty principle sets a limit to our capacity to predict the outcomes of two incompatible measurements. The Heisenberg uncertainty relation for two arbitrary observables A and B is usually discussed in textbooks. However, little or no attention is paid to the fact that Schrödinger generalised the Heisenberg relation taking into account the covariance between the observables A and B . This extended inequality is known as the Robertson–Schrödinger uncertainty relation. Here, we demonstrate the less known fact that two-level quantum states, i.e., qubits, satisfy the equality of the Robertson–Schrödinger uncertainty relation for two arbitrary observables A and B . Taking advantage of the homomorphism between SU(2) and SO(3) groups, it is possible to map the distributions of the expectation values and variances of the observables, and the Heisenberg and covariance terms on the Bloch sphere. The graphical visualisation of the relevant quantities involved in the uncertainty relations allows us to distinguish specific properties and symmetries that are not so evident in the algebraic formalism.
Abstract We have already mentioned that Schriöidinger’s formulation of quantum mechanics was slightly preceded by a totally different algebraic formulation proposed by Heisenberg. Although Schriöidinger’s methods were so much simpler that they were generally preferred, both approaches seemed to give the same answers, so it was natural to ask how they were related. This was particularly important because some features of quantum theory were easier to understand in Heisenberg’s formalism. In fact, as we have already noted, Schriöidinger was able to show that the two methods were essentially equivalent. Soon mathematicians like Hermann Weyland John von Neumann were able to find a general mathematical scheme that encompassed both approaches, and it is this that we shall now describe.