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

Chinese Physics Letters

Geometric Phase in Path Integral and Quantisation Rule

Zheng-Chuan Wang, Bo‐Zang Li

Published Apr 1, 1999• 2 citations
Publisher

Abstract

A description of geometric phase in terms of path integral formalism is presented. It is proved that this adiabatic phase can appear in the propagator or Green function of an adiabatic system. In the semiclassical approximation, following the Green function expression of the electronic density of states, the corresponding generalized Bohr-Sommerfeld quantisation rule can thus be obtained. It is shown that this rule has been corrected by the geometric phase.

DOI
10.1088/0256-307x/16/4/003

Related papers

Semiclassical approximation in the Weyl picture by path summation

T B Smith

Quantum mechanics may be formulated in pseudo-phasespace in the Weyl picture. The semiclassical expression for the Weyl transform of the usual time evolution operator is evaluated by summing over paths a semiclassical approximation for a new propagator in pseudo-phasespace. As an example the Bohr-Sommerfeld quantisation rule is recovered. The new propagator may in principle be used to construct other interesting quantities, such as the propagator governing the time evolution of the Wigner function.

Journal of Physics A Mathematical and General•Nov 1, 1978•20 citations

Adiabatic Approximation for Path Integrals and Geometrical Potentials

Takashi Ohnishi, Naoto Nagaosa

Geometrical concepts have played an important role in quantum mechanics. In the Schrodinger representation, the Berry–Simon connection and the quantum geometric tensor appear as gauge potentials in the effective Hamiltonian for a heavy system under the Born–Oppenheimer approximation. On the other hand, the Berry phase should arise as the amplitude of each path in the pathintegral formalism. In their pioneering work, Kuratsuji and Iida have developed the treatment of the Born– Oppenheimer approximation in the path integral quantization, where the Berry phase of the light system naturally appears in the effective action for the heavy system. However, they did not derive the gauge potentials themselves in the effective Hamiltonian. In this short note, we show that the gauge potentials which relate to the geometry of the Hilbert space of the light system are also derived in the adiabatic scheme of path integrals. Our result presents the path-integral formalism to consider the effective dynamics of the heavy system under the adiabatic approximation. Let us consider the following Hamiltonian:

Journal of the Physical Society of Japan•Dec 25, 2006•2 citations

Dynamical Meaning of Quantum Adiabatic Phase: The Case of a Noncanonical System

Hiroshi Kuratsuji, Shinji Iida

A dynamical meaning is given for the phase $\ensuremath{\Gamma}$ recently found in a quantum adiabatic process. A noncanonical system, specifically a spin system coupled with a proper intrinsic system, is considered by use of a path integral in a SU (2) coherent-state representation. It is shown that $\ensuremath{\Gamma}$ appears as an addition of topological nature to the conventional action function, which leads to a novel form of the semiclassical quantization rule including the phase $\ensuremath{\Gamma}$.

Physical Review Letters•Mar 10, 1986•41 citations

The Bohr-Sommerfeld quantization rule and the Weyl correspondence

Petros N. Argyres

With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.

Physics Physique Fizika•Nov 1, 1965•30 citations

Cyclic Evolution in Quantum Mechanics and the Phases of Bohr-Sommerfeld and Maslov

Robert G. Littlejohn

The Bohr-Sommerfeld and Maslov phases, which occur in the Einstein-Brillouin-Keller quantization rule of semiclassical mechanics, are shown to be examples of Berry's phase. The generalization of Berry's phase due to Aharonov and Anandan, which does not require adiabaticity, is employed.

Physical Review Letters•Nov 7, 1988•39 citations

The phase formalism for the one-dimensional eigenvalue problem and its relation with the quantum Bohr-Sommerfeld rule

A. V. Abarenov, A. V. Stolyarov

A quantum phase formalism is offered for searching for eigenvalues of the one-dimensional Schrodinger equation. Its application in the shooting method appreciably sped up convergence and made it absolute. The introduced quantum phase functions are close to their semiclassical analogues, unlike those used before, and transform into them in the semiclassical limit. A form of quantum analogue of the Bohr-Sommerfeld quantisation rule, allowing it to operate with non-integer quantum numbers within the framework of precise quantum consideration is obtained.

Journal of Physics B Atomic Molecular and Optical Physics•Aug 14, 1990•29 citations
A description of geometric phase in terms of path integral formalism is presented. It is proved that this adiabatic phase can appear in the propagator or Green function of an adiabatic system. In the semiclassical approximation, following the Green function expression of the electronic density of states, the corresponding generalized Bohr-Sommerfeld quantisation rule can thus be obtained. It is shown that this rule has been corrected by the geometric phase.
Quantum mechanics may be formulated in pseudo-phasespace in the Weyl picture. The semiclassical expression for the Weyl transform of the usual time evolution operator is evaluated by summing over paths a semiclassical approximation for a new propagator in pseudo-phasespace. As an example the Bohr-Sommerfeld quantisation rule is recovered. The new propagator may in principle be used to construct other interesting quantities, such as the propagator governing the time evolution of the Wigner function.
Geometrical concepts have played an important role in quantum mechanics. In the Schrodinger representation, the Berry–Simon connection and the quantum geometric tensor appear as gauge potentials in the effective Hamiltonian for a heavy system under the Born–Oppenheimer approximation. On the other hand, the Berry phase should arise as the amplitude of each path in the pathintegral formalism. In their pioneering work, Kuratsuji and Iida have developed the treatment of the Born– Oppenheimer approximation in the path integral quantization, where the Berry phase of the light system naturally appears in the effective action for the heavy system. However, they did not derive the gauge potentials themselves in the effective Hamiltonian. In this short note, we show that the gauge potentials which relate to the geometry of the Hilbert space of the light system are also derived in the adiabatic scheme of path integrals. Our result presents the path-integral formalism to consider the effective dynamics of the heavy system under the adiabatic approximation. Let us consider the following Hamiltonian:
A dynamical meaning is given for the phase $\ensuremath{\Gamma}$ recently found in a quantum adiabatic process. A noncanonical system, specifically a spin system coupled with a proper intrinsic system, is considered by use of a path integral in a SU (2) coherent-state representation. It is shown that $\ensuremath{\Gamma}$ appears as an addition of topological nature to the conventional action function, which leads to a novel form of the semiclassical quantization rule including the phase $\ensuremath{\Gamma}$.
With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.
A quantum phase formalism is offered for searching for eigenvalues of the one-dimensional Schrodinger equation. Its application in the shooting method appreciably sped up convergence and made it absolute. The introduced quantum phase functions are close to their semiclassical analogues, unlike those used before, and transform into them in the semiclassical limit. A form of quantum analogue of the Bohr-Sommerfeld quantisation rule, allowing it to operate with non-integer quantum numbers within the framework of precise quantum consideration is obtained.