Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Problem of the time operator and collision duration

V.S. Ol'khovskii

Published Nov 1, 1973• 2 citations
OpenAlex

Abstract

The notion of collision duration and properties of the symmetrical time operator in the nonrelativistic quantum mechanics are investigated. Using the theory of genenalized extensions of symmetrical operators, a conclusion is made: not only self-adjoint but also maximal symmetrical operators can correspond to the observables in the quantum mechanics. (auth)

Related papers

TIME AS A QUANTUM OBSERVABLE

V. S. Olkhovsky, Erasmo Recami

Some new results are presented and recent developments are reviewed on the study of Time in quantum physics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the Hermitian (more precisely, maximal Hermitian, but non-self-adjoint) operator for Time which appears: (i) for particles, in ordinary nonrelativistic quantum mechanics; and (ii) for photons (i.e. in first-quantization quantum electrodynamics). In conclusion, various recent and possible future applications of the time quantum analysis for tunnelling processes, nuclear collisions and systems with (quasi)discrete energy spectra are indicated.

arXiv:quant-ph/0605069•International Journal of Modern Physics A•Nov 10, 2007•39 citations

Non-self-adjoint operators as observables in quantum theory and nuclear physics

V. S. Olkhovsky, Sergei P. Maydanyuk, Erasmo Recami

The aim of this paper is to show the importance and applicability of non-self-adjoint operators for description of observables both in relativistic and nonrelativistic quantum mechanics, as well as in quantum electrodynamics. The paper considers (i) a maximally Hermitian (but non-self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics and (ii) the problem of defining the four-dimensional coordinate and momentum operators describing relativistic particles with zero spin (each of these operators contains both Hermitian and anti-Hermitian parts). Some other applications of non-selfadjoint and non-Hermitian operators in physics are analyzed as well. The paper ends with an analysis of a few cases of T -non-invariant interactions, including quantum dissipation and nuclear optical potentials.

Physics of Particles and Nuclei•Jul 1, 2010•6 citations

ON NON-SELF-ADJOINT OPERATORS FOR OBSERVABLES IN QUANTUM MECHANICS AND QUANTUM FIELD THEORY

Erasmo Recami, V. S. Olkhovsky, Sergei P. Maydanyuk

The aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.

arXiv:0903.3187•International Journal of Modern Physics A•Apr 10, 2010•14 citations

Observables in Quantum Mechanics and the Importance of Self-Adjointness

Tajron Jurić

We are focused on the idea that observables in quantum physics are a bit more then just hermitian operators and that this is, in general, a “tricky business”. The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.

arXiv:2103.01080•Universe•Feb 17, 2022•15 citations

Observables, maximal symmetric operators, POV measures and their applications in quantum mechanics

Robert H. Fountain

Approximate observables, their description in terms of POV measures and \nthe subsequent extension of the orthodox notion of observable are examined. \nThe sense in which the approximate observables are non-ideal observables \nis considered and a generalised concept of ideal observable is proposed. The \nconventional requirement of self-adjointness is relaxed to one of maximal \nsymmetry and the PV measures are replaced by the larger set of generalised \nspectral measures of maximal symmetric operators. The implications for \nthe quantisation problem are discussed and we give some new examples of \nobservables which have no orthodox representation, i.e. observables which \ncan be identified with non-self-adjoint maximal symmetric operators. Particular \nattention is paid to those aspects of the orthodox theory which rely \non specific properties of self-adjoint operators, such as their association with \nthe infinitesimal generators of groups of unitary operators, and we look at \nthe analogous properties in our generalised theory.

•Jan 1, 1996•0 citations

Self-adjointness and spontaneously broken symmetry

ANTON Z. CAPRI

The notion of self-adjointness as distinct from Hermiticity is usually avoided in elementary quantum mechanics courses due to a feeling that it has nothing to do with physics or else is too mathematical. This is definitely not the case, and we use the model of a free particle restricted to motion on a line over a finite interval, to demonstrate this. For such a particle, there is one-parameter infinity of self-adjoint extensions of the momentum operator. We use one of these, corresponding to ’’antiperiodic’’ boundary conditions, to show that physical phenomena different from the usual case (periodic boundary conditions) occur. In fact, in this case, we have a model displaying many of the properties of spontaneously broken symmetry.

American Journal of Physics•Sep 1, 1977•28 citations
The notion of collision duration and properties of the symmetrical time operator in the nonrelativistic quantum mechanics are investigated. Using the theory of genenalized extensions of symmetrical operators, a conclusion is made: not only self-adjoint but also maximal symmetrical operators can correspond to the observables in the quantum mechanics. (auth)
Some new results are presented and recent developments are reviewed on the study of Time in quantum physics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the Hermitian (more precisely, maximal Hermitian, but non-self-adjoint) operator for Time which appears: (i) for particles, in ordinary nonrelativistic quantum mechanics; and (ii) for photons (i.e. in first-quantization quantum electrodynamics). In conclusion, various recent and possible future applications of the time quantum analysis for tunnelling processes, nuclear collisions and systems with (quasi)discrete energy spectra are indicated.
The aim of this paper is to show the importance and applicability of non-self-adjoint operators for description of observables both in relativistic and nonrelativistic quantum mechanics, as well as in quantum electrodynamics. The paper considers (i) a maximally Hermitian (but non-self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics and (ii) the problem of defining the four-dimensional coordinate and momentum operators describing relativistic particles with zero spin (each of these operators contains both Hermitian and anti-Hermitian parts). Some other applications of non-selfadjoint and non-Hermitian operators in physics are analyzed as well. The paper ends with an analysis of a few cases of T -non-invariant interactions, including quantum dissipation and nuclear optical potentials.
The aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.
We are focused on the idea that observables in quantum physics are a bit more then just hermitian operators and that this is, in general, a “tricky business”. The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.
Approximate observables, their description in terms of POV measures and \nthe subsequent extension of the orthodox notion of observable are examined. \nThe sense in which the approximate observables are non-ideal observables \nis considered and a generalised concept of ideal observable is proposed. The \nconventional requirement of self-adjointness is relaxed to one of maximal \nsymmetry and the PV measures are replaced by the larger set of generalised \nspectral measures of maximal symmetric operators. The implications for \nthe quantisation problem are discussed and we give some new examples of \nobservables which have no orthodox representation, i.e. observables which \ncan be identified with non-self-adjoint maximal symmetric operators. Particular \nattention is paid to those aspects of the orthodox theory which rely \non specific properties of self-adjoint operators, such as their association with \nthe infinitesimal generators of groups of unitary operators, and we look at \nthe analogous properties in our generalised theory.
The notion of self-adjointness as distinct from Hermiticity is usually avoided in elementary quantum mechanics courses due to a feeling that it has nothing to do with physics or else is too mathematical. This is definitely not the case, and we use the model of a free particle restricted to motion on a line over a finite interval, to demonstrate this. For such a particle, there is one-parameter infinity of self-adjoint extensions of the momentum operator. We use one of these, corresponding to ’’antiperiodic’’ boundary conditions, to show that physical phenomena different from the usual case (periodic boundary conditions) occur. In fact, in this case, we have a model displaying many of the properties of spontaneously broken symmetry.