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College Physics

Discussion on the existence of common eigenstates for any two operators

Wan-Yu Ye

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

The general representation of uncertainty principle for any two quantum operators is defined,and the prerequisite of the existence of common eigenvector for any two operators with applications to several non-Hermitian cases is presented

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Simultaneous observables

Efstratios Manousakis

Abstract This chapter discusses when two observables, which are represented by operators, –can be simultaneously observable. The chapter discusses the uncertainty principle in the general case and when two operators can share eigenstates. It also discuss the consequences of symmetry.

•Nov 1, 2015•0 citations

THE PROBLEM OF THE TWO-MODE PHASE OPERATORS

LI Hong-cai, 福建师范大学物理系

The Hermitian cosine and sine operators for two particular models of two-mode space are obtained. As they commute, the common eigenstates are constructed. The properties of the operators and eigenvectors are discussed. The expectation values of these operators under the P-B phase representation are also studied.

Acta Physica Sinica•Jan 1, 1997•0 citations

Uncertainty Relations of Non-Hermitian Operators: Theory and Experimental Scheme

Xinzhi Zhao, Chengjie Zhang

The theoretical framework for the uncertainty relation of Hermitian operators is perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in the experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalized this unitary uncertainty relation theory and proposed uncertainty relations of non-Hermitian operators. Due to the difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplified the uncertainty relation of two non-Hermitian operators with pure states and proposed a realizable experimental measurement scheme by using the Mach–Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtained a generalized and analogous result of theirs.

Frontiers in Physics•Apr 19, 2022•9 citations

Minimum-uncertainty states for noncanonical operators

Ravinder Rupchand Puri

The usual uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}〈C${\mathrm{〉}}^{2}$/4 between two Hermitian operators A and B satisfying the noncanonical commutation relation [A,B]=iC, where C is not a constant multiple of the unit operator, fails to give a nontrivial lower bound on the product of the variances \ensuremath{\Delta}${\mathit{A}}^{2}$ and \ensuremath{\Delta}${\mathit{B}}^{2}$ when 〈C〉=0. For those operators, therefore, the general uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}[〈C${\mathrm{〉}}^{2}$+〈F${\mathrm{〉}}^{2}$]/4 where 〈F〉=〈AB+BA〉-2〈A〉〈B〉 is better suited to determine the lower bound on \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$. The implications of the general uncertainty relation and the properties of the minimum-uncertainty states, i.e., the states for which the general uncertainty relation is satisfied with equality, are discussed. The minimum-uncertainty states are found to fall in two different classes. One is the usually studied class for which 〈F〉=0, that is, the case when the usual uncertainty relation holds. The other is the hitherto unnoticed class for 〈C〉=0, that is, when the usual uncertainty relation is redundant. The squeezing properties of the present class of minimum-uncertainty states is discussed by defining squeezing in the light of the general uncertainty relation.

Physical Review A•Mar 1, 1994•35 citations

Uncertainty Principle and Normal Operators—Some Comments

Wesley E. Brittin

It is shown that an appropriate formulation of the uncertainty principle for normal operators leads to the same mathematical expression as that for Hermitian operators. The treatment is valid for mixed states as well as for pure states.

American Journal of Physics•Oct 1, 1966•2 citations

Non-divergent representation of a non-Hermitian operator near the exceptional point with application to a quantum Lorentz gas

K. Hashimoto, K. Kanki, H. Hayakawa +1 more

We propose a non-singular representation for a non-Hermitian operator even if the parameter space contains exceptional points (EPs), at which the operator cannot be diagonalized and the usual spectral representation ceases to exist. Our representation has a generalized Jordan block form and is written in terms of extended pseudo-eigenstates. Our method is free from divergence in the spectral representation at EPs, at which multiple eigenvalues and eigenvectors coalesce and the eigenvectors cannot be normalized. Our representation improves the accuracy of numerical calculations of physical quantities near EPs. We also find that our method is applicable to various problems related to EPs in the parameter space of non-Hermitian operators. We demonstrate the usefulness of our representation by investigating Boltzmann's collision operator in a 1D quantum Lorentz gas in the weak-coupling approximation.

arXiv:1409.7453•Progress of Theoretical and Experimental Physics•Feb 6, 2015•15 citations
The theoretical framework for the uncertainty relation of Hermitian operators is perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in the experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalized this unitary uncertainty relation theory and proposed uncertainty relations of non-Hermitian operators. Due to the difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplified the uncertainty relation of two non-Hermitian operators with pure states and proposed a realizable experimental measurement scheme by using the Mach–Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtained a generalized and analogous result of theirs.
The usual uncertainty relation \ensuremath{\Delta}A2{\mathit{A}}^{2}A2\ensuremath{\Delta}B2{\mathit{B}}^{2}B2\ensuremath{\ge}〈C〉2{\mathrm{〉}}^{2}〉2/4 between two Hermitian operators A and B satisfying the noncanonical commutation relation [A,B]=iC, where C is not a constant multiple of the unit operator, fails to give a nontrivial lower bound on the product of the variances \ensuremath{\Delta}A2{\mathit{A}}^{2}A2 and \ensuremath{\Delta}B2{\mathit{B}}^{2}B2 when 〈C〉=0. For those operators, therefore, the general uncertainty relation \ensuremath{\Delta}A2{\mathit{A}}^{2}A2\ensuremath{\Delta}B2{\mathit{B}}^{2}B2\ensuremath{\ge}[〈C〉2{\mathrm{〉}}^{2}〉2+〈F〉2{\mathrm{〉}}^{2}〉2]/4 where 〈F〉=〈AB+BA〉-2〈A〉〈B〉 is better suited to determine the lower bound on \ensuremath{\Delta}A2{\mathit{A}}^{2}A2\ensuremath{\Delta}B2{\mathit{B}}^{2}B2. The implications of the general uncertainty relation and the properties of the minimum-uncertainty states, i.e., the states for which the general uncertainty relation is satisfied with equality, are discussed. The minimum-uncertainty states are found to fall in two different classes. One is the usually studied class for which 〈F〉=0, that is, the case when the usual uncertainty relation holds. The other is the hitherto unnoticed class for 〈C〉=0, that is, when the usual uncertainty relation is redundant. The squeezing properties of the present class of minimum-uncertainty states is discussed by defining squeezing in the light of the general uncertainty relation.
We propose a non-singular representation for a non-Hermitian operator even if the parameter space contains exceptional points (EPs), at which the operator cannot be diagonalized and the usual spectral representation ceases to exist. Our representation has a generalized Jordan block form and is written in terms of extended pseudo-eigenstates. Our method is free from divergence in the spectral representation at EPs, at which multiple eigenvalues and eigenvectors coalesce and the eigenvectors cannot be normalized. Our representation improves the accuracy of numerical calculations of physical quantities near EPs. We also find that our method is applicable to various problems related to EPs in the parameter space of non-Hermitian operators. We demonstrate the usefulness of our representation by investigating Boltzmann's collision operator in a 1D quantum Lorentz gas in the weak-coupling approximation.