A semiclassical connection is established between quantal and classical properties of a system whose Hamiltonian is slowly cycled by varying its parameters round a circuit. The quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n=(n l ); the classical property is a shift Delta theta l (I) in the lth angle variable for motion round a phase-space torus with actions I=(I l ); the connection is Delta theta l =- delta gamma / delta n l . Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of q 2 , qp and p 2 ; and the rotated rotator, consisting of a particle sliding freely round a non-circular hoop slowly turned round once in its own plane.A semiclassical connection is established between quantal and classical properties of a system whose Hamiltonian is slowly cycled by varying its parameters round a circuit. The quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n=(n l ); the classical property is a shift Delta theta l (I) in the lth angle variable for motion round a phase-space torus with actions I=(I l ); the connection is Delta theta l =- delta gamma / delta n l . Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of q 2 , qp and p 2 ; and the rotated rotator, consisting of a particle sliding freely round a non-circular hoop slowly turned round once in its own plane.

Journal of Physics A Mathematical and GeneralJan 11, 1985415 citations

2. Quanta

Philip Ball

Cairn.infoJan 1, 20190 citations

A unified description of classical and quantal scattering

Ralph Eric Turner

A unified description of classical and quantal scattering is presented. This description is unified both in the mathematical language and in the conceptual pictures that are used. Statistical states and observables, represented in phase space, are emphasized while both the time dependent trajectory and stationary state beam pictures are used. Semiclassical-type approximation schemes to the generalized differential cross section are presented for the single and double potential cases. Connection is made with the Born and distorted wave Born approximations, for both quantum and classical mechanics. A similar semiclassical-type approximation scheme for the double potential internal state kinetic cross section is presented. Connection is also made with the 'constant acceleration approximation' of Oppenheim and Bloom. Zwanzig's projection operator method is used to express the kinetic cross sections in terms of memory effects for the interacting observables.A unified description of classical and quantal scattering is presented. This description is unified both in the mathematical language and in the conceptual pictures that are used. Statistical states and observables, represented in phase space, are emphasized while both the time dependent trajectory and stationary state beam pictures are used. Semiclassical-type approximation schemes to the generalized differential cross section are presented for the single and double potential cases. Connection is made with the Born and distorted wave Born approximations, for both quantum and classical mechanics. A similar semiclassical-type approximation scheme for the double potential internal state kinetic cross section is presented. Connection is also made with the 'constant acceleration approximation' of Oppenheim and Bloom. Zwanzig's projection operator method is used to express the kinetic cross sections in terms of memory effects for the interacting observables.

Jan 1, 20100 citations

Quantal Bits

Herbert S. Green

Springer eBooksJan 1, 20000 citations

Nonadiabatic geometric phase in periodic linear systems

Y. C. Ge, Mark Sheard Child

The method used by Garrison and Wright for the calculation of complex geometrical phases in quantal dissipative systems is extended to all time periodic linear systems, including classical linear systems with nonunitary time evolution. Each eigenvector $|{x}_{i}(t)〉$ of the period map modified by a complex scaling exponent ${\ensuremath{\theta}}_{i}(t)$ has an associated complex geometrical phase, which admits a solid angle interpretation in two-dimensional systems. Pairs of the $|{x}_{i}(t)〉$ commonly determine quadratic periodic invariants corresponding to topologically distinct surfaces of the invariant torus, and a connection between the scaling exponents and classical angle variables is established. The method is illustrated by the calculation of explicit expressions for the classical and quantal nonadiabatic phases associated with angular momentum precession. A second application to a general quadratic Hamiltonian extends existing adiabatic results to the classical and quantal nonadiabatic phases of both elliptic and hyberbolic systems. The relevance of torus ensemble averaging in a classical angle-action formulation is discussed.The method used by Garrison and Wright for the calculation of complex geometrical phases in quantal dissipative systems is extended to all time periodic linear systems, including classical linear systems with nonunitary time evolution. Each eigenvector $|{x}_{i}(t)〉$ of the period map modified by a complex scaling exponent ${\ensuremath{\theta}}_{i}(t)$ has an associated complex geometrical phase, which admits a solid angle interpretation in two-dimensional systems. Pairs of the $|{x}_{i}(t)〉$ commonly determine quadratic periodic invariants corresponding to topologically distinct surfaces of the invariant torus, and a connection between the scaling exponents and classical angle variables is established. The method is illustrated by the calculation of explicit expressions for the classical and quantal nonadiabatic phases associated with angular momentum precession. A second application to a general quadratic Hamiltonian extends existing adiabatic results to the classical and quantal nonadiabatic phases of both elliptic and hyberbolic systems. The relevance of torus ensemble averaging in a classical angle-action formulation is discussed.

Physical Review AAug 1, 199818 citations

|{x}_{i}(t)〉

of the period map modified by a complex scaling exponent ${\ensuremath{\theta}}_{i}(t)$ has an associated complex geometrical phase, which admits a solid angle interpretation in two-dimensional systems. Pairs of the

|{x}_{i}(t)〉

commonly determine quadratic periodic invariants corresponding to topologically distinct surfaces of the invariant torus, and a connection between the scaling exponents and classical angle variables is established. The method is illustrated by the calculation of explicit expressions for the classical and quantal nonadiabatic phases associated with angular momentum precession. A second application to a general quadratic Hamiltonian extends existing adiabatic results to the classical and quantal nonadiabatic phases of both elliptic and hyberbolic systems. The relevance of torus ensemble averaging in a classical angle-action formulation is discussed.