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.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 PhysicsDec 15, 197957 citations

Quantum action-angle-variable analysis of basic systems

Robert A. Leacock, Michael J. Padgett

Quantum action-angle variables are used to describe and analyze a number of familiar systems. For a given system, the quantum canonical transformation from the old coordinates, e.g., linear or polar, to the new coordinates, action-angle variables, is found by generalizing the corresponding classical transformation using a method based upon the correspondence principle, the Hermiticity and canonical nature of the old coordinates, and the requirement that the Hamiltonian be independent of the quantum angle variable. The bound-state energy levels and other important system properties follow immediately from the canonical transformation. Harmonic oscillators of various dimensions and the three-dimensional angular momentum system are used as illustrations; these illustrations provide interesting alternatives to the usual quantum treatments.Quantum action-angle variables are used to describe and analyze a number of familiar systems. For a given system, the quantum canonical transformation from the old coordinates, e.g., linear or polar, to the new coordinates, action-angle variables, is found by generalizing the corresponding classical transformation using a method based upon the correspondence principle, the Hermiticity and canonical nature of the old coordinates, and the requirement that the Hamiltonian be independent of the quantum angle variable. The bound-state energy levels and other important system properties follow immediately from the canonical transformation. Harmonic oscillators of various dimensions and the three-dimensional angular momentum system are used as illustrations; these illustrations provide interesting alternatives to the usual quantum treatments.

American Journal of PhysicsMar 1, 198712 citations

Action-angle variables in quantum mechanics

Avijit Lahiri, Gautam Kumar Ghosh, T. K. Kar

Physics Letters AFeb 1, 19985 citations

Canonical transformations to action and angle variables and their representations in quantum mechanics

M. Moshińsky, T. H. Seligman

Annals of PhysicsSep 1, 197878 citations

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.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 PhysicsApr 1, 19681,108 citations

{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.