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Collection of Czechoslovak Chemical Communications

Two uncertainty relations

Lubomír Skála, V. Kapsa

Published Jan 1, 2011• 2 citations
Publisher

Abstract

Heisenberg and Robertson–Schrödinger uncertainty relations for the coordinate and momentum follow from two stronger uncertainty relations. The first uncertainty relation has classical character and its right-hand side can have an arbitrary value greater than or equal to zero. The second uncertainty relation has quantum character and its right-hand side equals h2/4; its existence is related to the existence of the envelop of the wave function. These two uncertainty relations cannot be obviously improved on. The equality sign in the second relation can be achieved for much larger class of the wave functions than in case of the Heisenberg or Robertson–Schrödinger uncertainty relations.

DOI
10.1135/cccc2011015

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Heisenberg and Robertson–Schrödinger uncertainty relations for the coordinate and momentum follow from two stronger uncertainty relations. The first uncertainty relation has classical character and its right-hand side can have an arbitrary value greater than or equal to zero. The second uncertainty relation has quantum character and its right-hand side equals h2/4; its existence is related to the existence of the envelop of the wave function. These two uncertainty relations cannot be obviously improved on. The equality sign in the second relation can be achieved for much larger class of the wave functions than in case of the Heisenberg or Robertson–Schrödinger uncertainty relations.
The historical development of uncertainty relation, begining with first Heisenberg's thoughts of uncertainty principle is summed up in this thesis. After proving validity of Schwarz inequality general uncertainty relation for two hermitian operators is obtained, and from this general version the validity of Heisenberg uncertainty relation is than proved. The most important part of this work is the obtention of two new uncertainty relations, which are stronger then Heisenberg or Robertson-Schrödinger uncertainty relation, and their specific form for two examples - a free particle in a state discribed by the gaussian wave packet and the linear harmonic oscillator with a wave function in the shape of gaussian packet.
It is known that the Heisenberg and Robertson-Schr\"odinger uncertainty relations can be replaced by sharper relations in which the ``classical'' (depending on the gradient of the phase of the wave function) and ``quantum'' (depending on the gradient of the envelope of the wave function) parts of the variances $\ensuremath{\langle}{(\ensuremath{\Delta}x)}^{2}\ensuremath{\rangle}$ and $\ensuremath{\langle}{(\ensuremath{\Delta}p)}^{2}\ensuremath{\rangle}$ are separated. In this paper, multidimensional generalization of these relations is discussed.
A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schrödinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.
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.
The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.
Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.