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

Physical Review Letters

Quantum Action-Angle Variables for the Harmonic Oscillator

Harry R. Lewis, W. E. Lawrence, Joseph David Harris

Published Dec 23, 1996• 27 citations
Publisher

Abstract

Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of $q$ and the eigenstates of $p$. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.

DOI
10.1103/physrevlett.77.5157

Related papers

Classical and quantum action-phase variables for time-dependent oscillators

Sang Pyo Kim, Don Nelson Page

For a time-dependent classical quadratic oscillator we introduce pairs of real and complex invariants that are linear in position and momentum. Each pair of invariants realize explicitly a canonical transformation from the phase space to the invariant space, in which the action-phase variables are defined. We find the action operator for the time-dependent quantum oscillator via the classical-quantum correspondence. Candidate phase operators conjugate to the action operator are discussed, but no satisfactory ones are found.

arXiv:quant-ph/0101076•Physical Review A•Jun 6, 2001•42 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 time-dependent perturbation theory for the construction of operator invariants

Harry R. Lewis, W. E. Lawrence

A time-dependent perturbation theory is presented for constructing operator invariants iteratively when a Hamiltonian consists of a time-independent zeroth-order term plus a time-dependent perturbation. The procedure uses a single canonical transformation that involves an invariant operator that is canonically conjugate to the zeroth-order Hamiltonian. Such an invariant operator has recently been constructed for the case in which the zeroth-order Hamiltonian describes a harmonic oscillator.

Physical Review A•Apr 1, 1997•6 citations

Canonical Operators for the Simple Harmonic Oscillator

Boris Leaf

Angle and action operators (w, j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q, k) generated by a one-sided unitary operator U such that U†U = 1 and UU† commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors |η〉 are constructed for every real value of η; the set {|η〉} is complete and orthogonal. Another complete set {|W〉} is obtained, consisting of the Fourier transforms of the kets in the set {|η〉}. The angle operator is w=U†qU=∫dW |W〉 W 〈W|. The set {|W〉} is not orthogonal; |W〉 is not an eigenvector of w. If v is defined as ∫dW|W〉exp (−2πiW) 〈W|, then the creation and destruction operators are given by a = vn½, a† = n½v†. v is a one-sided unitary operator such that vv† = 1, but v†v = 1 − |0〉 〈0|, where |0〉 is the ground state of the oscillator; v and v† are similar to the operators E- and E+ of Carruthers and Nieto. The Weyl transforms of w and j = 2πℏ(n + ½1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.

Journal of Mathematical Physics•Nov 1, 1969•26 citations

Action-angle variables in quantum mechanics

Stuart D. Augustin, HERSCHEL A. RABITZ

Conventional quantum mechanical treatments of many systems have worked with coordinates and momenta that are not canonically conjugate. In this work it is shown how the quantum expressions may be reformulated in terms of the canonical set of action-angle variables, and specific examples of the harmonic oscillator, linear rotor, and triaxial rotor are presented. When expressed in these terms, the quantum mechanics take on a form which can be directly related to analogous results from classical mechanics. In addition, it becomes possible to express the Hamiltonian in the minimum number of coordinates. It is also shown that the common assumption of an exponential form for the overlap of canonical coordinate and momentum eigenstates is false for an asymmetric rotor. This has important implications for the quantization rules applicable to nonseparable systems.

The Journal of Chemical Physics•Dec 15, 1979•57 citations

Quantization in terms of symplectic groups: The harmonic oscillator as a generic example

H.A. Kastrup

The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.

Journal of Physics Conference Series•Mar 1, 2011•1 citation
Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of qqq and the eigenstates of ppp. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.
For a time-dependent classical quadratic oscillator we introduce pairs of real and complex invariants that are linear in position and momentum. Each pair of invariants realize explicitly a canonical transformation from the phase space to the invariant space, in which the action-phase variables are defined. We find the action operator for the time-dependent quantum oscillator via the classical-quantum correspondence. Candidate phase operators conjugate to the action operator are discussed, but no satisfactory ones are found.
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.
Angle and action operators (w, j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q, k) generated by a one-sided unitary operator U such that U†U = 1 and UU† commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors |η〉 are constructed for every real value of η; the set {|η〉} is complete and orthogonal. Another complete set {|W〉} is obtained, consisting of the Fourier transforms of the kets in the set {|η〉}. The angle operator is w=U†qU=∫dW |W〉 W 〈W|. The set {|W〉} is not orthogonal; |W〉 is not an eigenvector of w. If v is defined as ∫dW|W〉exp (−2πiW) 〈W|, then the creation and destruction operators are given by a = vn½, a† = n½v†. v is a one-sided unitary operator such that vv† = 1, but v†v = 1 − |0〉 〈0|, where |0〉 is the ground state of the oscillator; v and v† are similar to the operators E- and E+ of Carruthers and Nieto. The Weyl transforms of w and j = 2πℏ(n + ½1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.
Conventional quantum mechanical treatments of many systems have worked with coordinates and momenta that are not canonically conjugate. In this work it is shown how the quantum expressions may be reformulated in terms of the canonical set of action-angle variables, and specific examples of the harmonic oscillator, linear rotor, and triaxial rotor are presented. When expressed in these terms, the quantum mechanics take on a form which can be directly related to analogous results from classical mechanics. In addition, it becomes possible to express the Hamiltonian in the minimum number of coordinates. It is also shown that the common assumption of an exponential form for the overlap of canonical coordinate and momentum eigenstates is false for an asymmetric rotor. This has important implications for the quantization rules applicable to nonseparable systems.
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.