A way of construction of semiclassical wave function (SWF) based on the Maslov - Fedoriuk approach is proposed which appears to be appropriate also for systems with chaotic classical limits. Some classical constructions called skeletons are considered. The skeletons are generalizations of Arnolds' tori able to gather chaotic dynamics. SWF's are continued by caustic singularities in the configuration space rather then in the phase space using complex time method. The skeleton formulation provides us with a new algorithm for the semiclassical approximation method which is applied to construct SWF's as well as to calculate energy spectra for the circular and rectangular billiards as well as to construct the simplest SWF's and the respective spectrum for the Bunimovich stadium. The scar phenomena are considered and a possibility of their description by the skeleton method is discussed. PACS number(s): 03.65.-w, 03.65.Sq, 02.30.Jr, 02.30.Lt, 02.30.Mv Key Words: Schrödinger equation, semiclassical expansion, Lagrange manifolds, classical trajectories, chaotic dynamics, quantum chaos, scarsA way of construction of semiclassical wave function (SWF) based on the Maslov - Fedoriuk approach is proposed which appears to be appropriate also for systems with chaotic classical limits. Some classical constructions called skeletons are considered. The skeletons are generalizations of Arnolds' tori able to gather chaotic dynamics. SWF's are continued by caustic singularities in the configuration space rather then in the phase space using complex time method. The skeleton formulation provides us with a new algorithm for the semiclassical approximation method which is applied to construct SWF's as well as to calculate energy spectra for the circular and rectangular billiards as well as to construct the simplest SWF's and the respective spectrum for the Bunimovich stadium. The scar phenomena are considered and a possibility of their description by the skeleton method is discussed. PACS number(s): 03.65.-w, 03.65.Sq, 02.30.Jr, 02.30.Lt, 02.30.Mv Key Words: Schrödinger equation, semiclassical expansion, Lagrange manifolds, classical trajectories, chaotic dynamics, quantum chaos, scars

arXiv:1108.2527Aug 11, 20115 citations

Method of the semiclassical representation of quantum theory

Eugeni P. Gordov, Stanislav D. Tvorogov

Jan 1, 19845 citations

Semiclassical Descriptions of Quantum Mechanics

WORLD SCIENTIFIC eBooksDec 1, 19980 citations

Semiclassical theory for the Maslov-type wave packet: Hierarchy below the semiclassical Feynman kernel

Atsuko Inoue-Ushiyama, Kazuo Takatsuka

A semiclassical theory based on the Maslov-type wave packet is reported. The present approximation constitutes a class of semiclassical theory that brings about a hierarchy of theoretical structures below or equivalent to the standard semiclassical approximation to the Feynman kernel. We first propose a semiclassical wave function; that is, what we call the action-decomposed function (ADF). An ADF is propagated in time semiclassically along a ``single action surface,'' which is characterized in terms of a given initial momentum, with the phase being proportional to the ${F}_{2}(q,p)$ -type generating function. An arbitrary wave function can be expanded continuously in the ADF's of different initial momenta, each of which is associated with weighting factors both in configuration and momentum spaces. Depending on the form of the weighting function, the present scheme covers a number of different levels of approximation, ranging from a ``single'' ADF to a semiclassical kernel. Thus our theory provides a way of connecting the Maslov-type semiclassical wave packet to the WKB theory based on the Feynman kernel. It is generally concluded that the above kernel limit attains the highest accuracy possible within the present scheme, but it requires a large number of classical trajectories to represent it. On the other hand, a single ADF yields a little less accurate results but demands the fewest representing classical trajectories due to the narrowest momentum distribution. In fact, we show numerically that the autocorrelation function represented in terms of a single ADF, from which the energy spectrum is extracted with a Fourier transformation, can be calculated with drastically fewer classical trajectories without losing much accuracy. We also present a limitation in applying the ADF, and show an interesting symptom arising from a pathological use of the approximation. The present theory is quite promising for a spectral analysis of the vibrational states of relatively large molecules, if applied appropriately.A semiclassical theory based on the Maslov-type wave packet is reported. The present approximation constitutes a class of semiclassical theory that brings about a hierarchy of theoretical structures below or equivalent to the standard semiclassical approximation to the Feynman kernel. We first propose a semiclassical wave function; that is, what we call the action-decomposed function (ADF). An ADF is propagated in time semiclassically along a ``single action surface,'' which is characterized in terms of a given initial momentum, with the phase being proportional to the ${F}_{2}(q,p)$-type generating function. An arbitrary wave function can be expanded continuously in the ADF's of different initial momenta, each of which is associated with weighting factors both in configuration and momentum spaces. Depending on the form of the weighting function, the present scheme covers a number of different levels of approximation, ranging from a ``single'' ADF to a semiclassical kernel. Thus our theory provides a way of connecting the Maslov-type semiclassical wave packet to the WKB theory based on the Feynman kernel. It is generally concluded that the above kernel limit attains the highest accuracy possible within the present scheme, but it requires a large number of classical trajectories to represent it. On the other hand, a single ADF yields a little less accurate results but demands the fewest representing classical trajectories due to the narrowest momentum distribution. In fact, we show numerically that the autocorrelation function represented in terms of a single ADF, from which the energy spectrum is extracted with a Fourier transformation, can be calculated with drastically fewer classical trajectories without losing much accuracy. We also present a limitation in applying the ADF, and show an interesting symptom arising from a pathological use of the approximation. The present theory is quite promising for a spectral analysis of the vibrational states of relatively large molecules, if applied appropriately.

Physical Review AMay 1, 199927 citations

Semiclassical wavefunctions and Schrodinger equation

Maurice A. de Gosson

CRC Press eBooksMar 6, 20030 citations

{F}_{2}(q,p)

-type generating function. An arbitrary wave function can be expanded continuously in the ADF's of different initial momenta, each of which is associated with weighting factors both in configuration and momentum spaces. Depending on the form of the weighting function, the present scheme covers a number of different levels of approximation, ranging from a ``single'' ADF to a semiclassical kernel. Thus our theory provides a way of connecting the Maslov-type semiclassical wave packet to the WKB theory based on the Feynman kernel. It is generally concluded that the above kernel limit attains the highest accuracy possible within the present scheme, but it requires a large number of classical trajectories to represent it. On the other hand, a single ADF yields a little less accurate results but demands the fewest representing classical trajectories due to the narrowest momentum distribution. In fact, we show numerically that the autocorrelation function represented in terms of a single ADF, from which the energy spectrum is extracted with a Fourier transformation, can be calculated with drastically fewer classical trajectories without losing much accuracy. We also present a limitation in applying the ADF, and show an interesting symptom arising from a pathological use of the approximation. The present theory is quite promising for a spectral analysis of the vibrational states of relatively large molecules, if applied appropriately.