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

Physics Physique Fizika

Quantum mechanical phase and time operator

Leonard Susskind, Jonathan Glogower

Published Jul 1, 1964• 812 citations
Publisher

Abstract

The phase operator for an oscillator is shown not to exist. It is replaced by a pair of non-commuting sin and cos operators which can be used to define uncertainty relations for phase and number. The relation between phase and angle operators is carefully discussed. The possibility of using a phase variable as a quantum clock is demonstrated and the states for which the clock is most accurate are constructed.

DOI
10.1103/physicsphysiquefizika.1.49

Related papers

Phase and Angle Variables in Quantum Mechanics

Peter Carruthers, Michael Martin Nieto

The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state vectors under consideration are clarified in the context of the Heisenberg uncertainty relations. The familiar case of the azimuthal angle variable $\ensuremath{\phi}$ and its "conjugate" angular momentum ${L}_{z}$ is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables ($sin\ensuremath{\phi}$ and $cos\ensuremath{\phi}$ to describe the phase variable. Well-defined uncertainty relations are derived and discussed. A detailed analysis of the three-dimensional harmonic oscillator excited in coherent states is given. A detailed analysis of the simple harmonic oscillator is given. The usual assumption that a (Hermitian) phase operator $\ensuremath{\varphi}$ (conjugate to the number operator $N$) exists is shown to be erroneous. However, cosine and sine operators $C$ and $S$ exist and are the appr\'opriate phase variables. A Poisson bracket argument using action-angle (rather $J$, $cos\ensuremath{\varphi}$, $sin\ensuremath{\varphi}$) variables is used to deduce $C$ and $S$. The spectra and eigenfunctions of these operators are investigated, along with the important "phase-difference" periodic variables. The properties of the oscillator variables in the various types of states are analyzed with special attention to the uncertainty relations and the transition to the classical limit. The utility of coherent states as a basis for the description of the evolution of the density matrix is emphasized. In this basis it is easy to identify the classical Liouville equation in action-angle variables along with quantum-mechanical "corrections." Mention is made of possible physical applications to superfluid systems.

Reviews of Modern Physics•Apr 1, 1968•1,108 citations

Relative Phase and Time States

Masataka Ohta

It is pointed out that, while Suskind and Glogower claim “The phase operator for an oscillator is shown not to exist.”, they actually showed absolute phase operator, which is physically meaningless, does not exist. It is also pointed out that, they showed countably infinitely many trigonometric relative (relative between two oscillators) phase states, which form complete basis and are, in a sense, best possible quantum “phase states”, exist. In addition, continuously infinitely many phase states are constructed at classical, that is, infinitely many quanta, limit. Similarly, relative time states and operator are constructed with two oscillators with different angular velocities.

•Jun 1, 2019•0 citations

Time operator for the quantum harmonic oscillator: resolution of an apparent paradox

Alex Granik, Harry R. Lewis

An apparent paradox is resolved that concerns the existence of time operators which have been derived for the quantum harmonic oscillator. There is an apparent paradox because, although a time operator is canonically conjugate to the Hamiltonian, it has been asserted that no operator exists that is canonically conjugate to the Hamiltonian. In order to resolve the apparent paradox, we work in a representation where the phase operator is diagonal. The boundary condition on wave functions is such that they be periodic in the phase variable, which is related to the (continuous) eigenvalue of the time operator. Matrix elements of the commutator of the time operator with the Hamiltonian involve the phase variable itself in addition to periodic functions of the phase variable. The Hamiltonian is not hermitian when operating in space that includes the phase variable itself. The apparent paradox is resolved when this non-hermeticity is taken into account correctly in the evaluation of matrix elements of the commutation relation.

arXiv:quant-ph/0006063•Jun 14, 2000•0 citations

Quantum phase: A different approach

Marco A. Tapia-Valerdi, I. Ramos-Prieto, Francisco Soto +2 more

It is generally thought that, in the same way that position and momentum are canonical variables in quantum mechanics, the number operator should have a canonical operator, i.e., there should be a possibility of defining a well-defined phase operator associated with it. From the point of view of the phase space, it becomes clearer as position and momentum uncertainties are well reflected. Indeed, the same occurs in the phase space for phase and intensity. In this contribution, we do a small survey on the possibilities of defining a phase operator via discrete Fourier operators and introduce some possible applications.

International Journal of Modern Physics B•Dec 6, 2025•0 citations

Phase of the quantum oscillator

H. S. Sharatchandra

Requirements of a conjugate operator are emphasized, especially in its role in uncertainty relations.It is argued that in many contexts it is necessary to extend the Hilbert space in order to define a conjugate operator as in gauge theories. Example of a particle in a box is analysed. This is closely related to the quantum oscillator through cosine states of Susskind and Glogower.It is used to justify London's phase wave functions albeit as part of a larger Hilbert space. A new definition phase uncertainty neccessiated by periodicity is proposed.It is close to the usual r.m.s. definition.Corresponding number- phase uncertainty relation is obtained and its implications are discussed. Hilbert space of an oscillator is identified with the Hilbert space of a planar rotor with a $Z_2$ gauge invariance.This is used to construct states analogous to the cosine and sine states and to illustrate unitary equivalence of Hilbert spaces.

arXiv:quant-ph/9710020•Oct 6, 1997•6 citations

Canonically Conjugate Pairs, Uncertainty Relations, and Phase Operators

J. C. Garrison, Jack Wong

Apparent difficulties that prevent the definition of canonical conjugates for certain observables, e.g., the number operator, are eliminated by distinguishing between the Heisenberg and Weyl forms of the canonical commutation relations (CCR's). Examples are given for which the uncertainty principle does not follow from the CCR's. An operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.

Journal of Mathematical Physics•Aug 1, 1970•155 citations
The phase operator for an oscillator is shown not to exist. It is replaced by a pair of non-commuting sin and cos operators which can be used to define uncertainty relations for phase and number. The relation between phase and angle operators is carefully discussed. The possibility of using a phase variable as a quantum clock is demonstrated and the states for which the clock is most accurate are constructed.
The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state vectors under consideration are clarified in the context of the Heisenberg uncertainty relations. The familiar case of the azimuthal angle variable $\ensuremath{\phi}$ and its "conjugate" angular momentum Lz{L}_{z}Lz​ is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables ($sin\ensuremath{\phi}$ and $cos\ensuremath{\phi}$ to describe the phase variable. Well-defined uncertainty relations are derived and discussed. A detailed analysis of the three-dimensional harmonic oscillator excited in coherent states is given. A detailed analysis of the simple harmonic oscillator is given. The usual assumption that a (Hermitian) phase operator $\ensuremath{\varphi}$ (conjugate to the number operator NNN) exists is shown to be erroneous. However, cosine and sine operators CCC and SSS exist and are the appr\'opriate phase variables. A Poisson bracket argument using action-angle (rather JJJ, $cos\ensuremath{\varphi}$, $sin\ensuremath{\varphi}$) variables is used to deduce CCC and SSS. The spectra and eigenfunctions of these operators are investigated, along with the important "phase-difference" periodic variables. The properties of the oscillator variables in the various types of states are analyzed with special attention to the uncertainty relations and the transition to the classical limit. The utility of coherent states as a basis for the description of the evolution of the density matrix is emphasized. In this basis it is easy to identify the classical Liouville equation in action-angle variables along with quantum-mechanical "corrections." Mention is made of possible physical applications to superfluid systems.
It is pointed out that, while Suskind and Glogower claim “The phase operator for an oscillator is shown not to exist.”, they actually showed absolute phase operator, which is physically meaningless, does not exist. It is also pointed out that, they showed countably infinitely many trigonometric relative (relative between two oscillators) phase states, which form complete basis and are, in a sense, best possible quantum “phase states”, exist. In addition, continuously infinitely many phase states are constructed at classical, that is, infinitely many quanta, limit. Similarly, relative time states and operator are constructed with two oscillators with different angular velocities.
An apparent paradox is resolved that concerns the existence of time operators which have been derived for the quantum harmonic oscillator. There is an apparent paradox because, although a time operator is canonically conjugate to the Hamiltonian, it has been asserted that no operator exists that is canonically conjugate to the Hamiltonian. In order to resolve the apparent paradox, we work in a representation where the phase operator is diagonal. The boundary condition on wave functions is such that they be periodic in the phase variable, which is related to the (continuous) eigenvalue of the time operator. Matrix elements of the commutator of the time operator with the Hamiltonian involve the phase variable itself in addition to periodic functions of the phase variable. The Hamiltonian is not hermitian when operating in space that includes the phase variable itself. The apparent paradox is resolved when this non-hermeticity is taken into account correctly in the evaluation of matrix elements of the commutation relation.
It is generally thought that, in the same way that position and momentum are canonical variables in quantum mechanics, the number operator should have a canonical operator, i.e., there should be a possibility of defining a well-defined phase operator associated with it. From the point of view of the phase space, it becomes clearer as position and momentum uncertainties are well reflected. Indeed, the same occurs in the phase space for phase and intensity. In this contribution, we do a small survey on the possibilities of defining a phase operator via discrete Fourier operators and introduce some possible applications.
Requirements of a conjugate operator are emphasized, especially in its role in uncertainty relations.It is argued that in many contexts it is necessary to extend the Hilbert space in order to define a conjugate operator as in gauge theories. Example of a particle in a box is analysed. This is closely related to the quantum oscillator through cosine states of Susskind and Glogower.It is used to justify London's phase wave functions albeit as part of a larger Hilbert space. A new definition phase uncertainty neccessiated by periodicity is proposed.It is close to the usual r.m.s. definition.Corresponding number- phase uncertainty relation is obtained and its implications are discussed. Hilbert space of an oscillator is identified with the Hilbert space of a planar rotor with a Z2Z_2Z2​ gauge invariance.This is used to construct states analogous to the cosine and sine states and to illustrate unitary equivalence of Hilbert spaces.