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 A

Semiquantized Hamiltonians in generalized coordinates

O. Zohni

Published May 1, 1981• 3 citations
Publisher

Abstract

A consistent prescription is given for constructing semiquantized Hamiltonians in generalized coordinates. The method extends the classical path approximation and deals directly with a generalized form of the quantized kinetic-energy operator to derive simplified quantum-mechanical wave equations for multidimensional dynamical systems exhibiting both quantal and classical features.

DOI
10.1103/physreva.23.2297

Related papers

Vector fields, line integrals, and the Hamilton-Jacobi equation: Semiclassical quantization of bound states

N. De Leon, Eric J. Heller

In 1972, P. Pechukas [J. Chem. Phys. 57, 5577 (1972)] proposed "classical states" associated with Miller-Good transformations, semiclassical quantization, and the Hamilton-Jacobi equation. In this paper we use some of the concepts established by Pechukas and extend them to define generalized curvilinear coordinates $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ associated with nonlinear dynamical systems with Hamiltonians of the form $H=T(\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}})+V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. For reasons discussed in the paper we call these coordinates "nodal" coordinates. We show that a transformation to nodal coordinates is to a very good approximation dependent only on Cartesian variables. Using this approximation we demonstrate that, within the regular regime of phase space, canonical transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates simplifies the analysis of the classical and quantum mechanics of dynamical systems. Our fundamental conclusions are as follows: (1) The solution of the Hamilton-Jacobi equation is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; (2) the WKB wave function ${\ensuremath{\Psi}}^{\mathrm{WKB}}$ is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; and (3) if the transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates is made conformal (which we show we are allowed to do within the approximation stated above), then the Schr\"odinger equation and wave functions become separable. Finally, the concepts in this paper are discussed for systems of two degrees of freedom but can be generalized to more degrees of freedom.

Physical Review A•Jul 1, 1984•41 citations

Statistical approach to quantum mechanics I: General nonrelativistic theory

G. H. Goedecke

In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schrödinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical Hamiltonian in the $N$-dimensional metric configuration space defined by the classical kinetic energy of the system, with the classical conjugate momentum $N$-vector replaced by $-i\hbar$ times the vector gradient operator in that space. We obtain these results by using conservation of probability, general tensor calculus, the Madelung transform, the Ehrenfest theorem and/or the Hamilton-Jacobi equation, and comparison with results for the charged harmonic oscillator in stochastic electrodynamics. We also provide two illustrative examples and a discussion of how coordinate trajectories could be compatible with wave properties such as interference, diffraction, and tunneling.

arXiv:1408.1722•Aug 7, 2014•0 citations

Hamiltonian Operator in Generalized Coordinates

Lionel D. Dureau

The quantum mechanical, time-dependent wave equation in generalized coordinates is obtained by a slight modification of Schrödinger's original derivation. If the classical Hamiltonian for a system is known, the variational technique may be applied to the Hamilton-Jacobi equation to yield an Euler-Lagrange equation. This latter is then the wave equation for the system. The method, which is generally easy to apply, is illustrated by applications to a single particle in curvilinear coordinates and to a symmetric top molecule.

American Journal of Physics•Nov 1, 1965•5 citations

Generalized coordinates in quantum mechanics

V. D. Krivchenkov

A prescription is proposed for the construction of quantum-mechanical operators on the basis of the classical Lagrangian function in generalized coordinates. Integrals of the motion and particular dynamical systems are considered.

Soviet Physics Uspekhi•Oct 31, 1981•2 citations

Unique Hamiltonian Operators via Feynman Path Integrals

Edward H. Kerner, William G. Sutcliffe

The old problem of how to represent uniquely a prescribed classical Hamiltonian H as a well-defined quantal operator Ĥ is shown to have a clear answer within Feynman's path-integral scheme (as expanded by Garrod) for quantum mechanics. The computation of Ĥ involves the momentum Fourier transform of a coordinate average of H. A differential equation for a reduced form of the Feynman propagator giving Ĥ from H is found; and the example of polynomial H worked out to give the Born-Jordan ordering rule for Ĥ in this case.

Journal of Mathematical Physics•Feb 1, 1970•36 citations

Hamiltonian Path-Integral Methods

Claude Garrod

A path-integral formulation of quantum mechanics is investigated which is closely related to that of Feynman. It differs from Feynman's formulation in that it involves the Hamiltonian function of the canonically conjugate coordinates and momenta. The classical limit yields the variational principle: $\ensuremath{\delta}ff(p\ifmmode\ddagger\else\textdaggerdbl\fi{}\ensuremath{-}N)dt=0$. A path-integral formula is also obtained for the energy eigenstate projection operator associated with the time-independent Schr\"odinger equation. The classical limit of the projection operator formula yields a modified form of the well-known variational principle for the phase-space orbit of given energy. Relativistically covariant Hamiltonian variational principles are analyzed and lead naturally to a relativistic scalar wave equation which involves a proper time variable which is canonically conjugate to the mass in the same manner as the ordinary time variable is conjugate to the energy in nonrelativistic quantum theory.

Reviews of Modern Physics•Jul 1, 1966•164 citations
A consistent prescription is given for constructing semiquantized Hamiltonians in generalized coordinates. The method extends the classical path approximation and deals directly with a generalized form of the quantized kinetic-energy operator to derive simplified quantum-mechanical wave equations for multidimensional dynamical systems exhibiting both quantal and classical features.
In 1972, P. Pechukas [J. Chem. Phys. 57, 5577 (1972)] proposed "classical states" associated with Miller-Good transformations, semiclassical quantization, and the Hamilton-Jacobi equation. In this paper we use some of the concepts established by Pechukas and extend them to define generalized curvilinear coordinates $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ associated with nonlinear dynamical systems with Hamiltonians of the form $H=T(\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}})+V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. For reasons discussed in the paper we call these coordinates "nodal" coordinates. We show that a transformation to nodal coordinates is to a very good approximation dependent only on Cartesian variables. Using this approximation we demonstrate that, within the regular regime of phase space, canonical transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates simplifies the analysis of the classical and quantum mechanics of dynamical systems. Our fundamental conclusions are as follows: (1) The solution of the Hamilton-Jacobi equation is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; (2) the WKB wave function ${\ensuremath{\Psi}}^{\mathrm{WKB}}$ is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; and (3) if the transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates is made conformal (which we show we are allowed to do within the approximation stated above), then the Schr\"odinger equation and wave functions become separable. Finally, the concepts in this paper are discussed for systems of two degrees of freedom but can be generalized to more degrees of freedom.
In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields the multi-coordinate Schrödinger equation, with its usual boundary conditions, as an essential statistical equation for the system. We derive the general "canonical quantization" rule, that the Hamiltonian operator must be the classical Hamiltonian in the NNN-dimensional metric configuration space defined by the classical kinetic energy of the system, with the classical conjugate momentum NNN-vector replaced by −iℏ-i\hbar−iℏ times the vector gradient operator in that space. We obtain these results by using conservation of probability, general tensor calculus, the Madelung transform, the Ehrenfest theorem and/or the Hamilton-Jacobi equation, and comparison with results for the charged harmonic oscillator in stochastic electrodynamics. We also provide two illustrative examples and a discussion of how coordinate trajectories could be compatible with wave properties such as interference, diffraction, and tunneling.
The quantum mechanical, time-dependent wave equation in generalized coordinates is obtained by a slight modification of Schrödinger's original derivation. If the classical Hamiltonian for a system is known, the variational technique may be applied to the Hamilton-Jacobi equation to yield an Euler-Lagrange equation. This latter is then the wave equation for the system. The method, which is generally easy to apply, is illustrated by applications to a single particle in curvilinear coordinates and to a symmetric top molecule.
The old problem of how to represent uniquely a prescribed classical Hamiltonian H as a well-defined quantal operator Ĥ is shown to have a clear answer within Feynman's path-integral scheme (as expanded by Garrod) for quantum mechanics. The computation of Ĥ involves the momentum Fourier transform of a coordinate average of H. A differential equation for a reduced form of the Feynman propagator giving Ĥ from H is found; and the example of polynomial H worked out to give the Born-Jordan ordering rule for Ĥ in this case.
A path-integral formulation of quantum mechanics is investigated which is closely related to that of Feynman. It differs from Feynman's formulation in that it involves the Hamiltonian function of the canonically conjugate coordinates and momenta. The classical limit yields the variational principle: $\ensuremath{\delta}ff(p\ifmmode\ddagger\else\textdaggerdbl\fi{}\ensuremath{-}N)dt=0$. A path-integral formula is also obtained for the energy eigenstate projection operator associated with the time-independent Schr\"odinger equation. The classical limit of the projection operator formula yields a modified form of the well-known variational principle for the phase-space orbit of given energy. Relativistically covariant Hamiltonian variational principles are analyzed and lead naturally to a relativistic scalar wave equation which involves a proper time variable which is canonically conjugate to the mass in the same manner as the ordinary time variable is conjugate to the energy in nonrelativistic quantum theory.