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

Quantum dynamics in dual spaces

E. C. G. Sudarshan

Published Sep 1, 1994• 11 citations
Publisher

Abstract

Quantum mechanics gives us information about the spectra of dynamical variables and transition rates including scattering cross sections. They can be exhibited as spectral information in analytically continued spaces and their duals. Quantum mechanics formulated in these generalized spaces is used to study scattering and time evolution. It is shown that the usual asymptotic condition is inadequate to deal with the scattering of composite or unstable particles. Scattering theory needs an amendment when the interacting system is not isospectral with the free Hamiltonian; the amendment is formulated. Perturbation theory in generalized spaces is developed and used to study the deletion and augmentation of the spectrum of the Hamiltonian. A complete set of algebraically independent constants for an interacting system is obtained. The question of the breaking of time symmetry is discussed.

DOI
10.1103/physreva.50.2006

Related papers

Quantum Mechanics in Fock Space

M. Moshińsky

In the present paper we propose to develop a quantum-mechanical scheme in Fock space that would describe interactions that take place through the formation of a compound particle. The discussion will be restricted to a dynamical system representing a single-level scattering process. The state of this dynamical system can be found in two stages: initial particles and compound nucleus. The dynamical behavior of this state will be given in terms of a hamiltonian (that has no classical analog) which will be derived from the boundary conditions satisfied by the state. The representation of the dynamical variables, associated with the two stages of the state, will be discussed, and a complete set of constants of motion of our dynamical system will be given. The expansion of an arbitarary Fock state in terms of the simultaneous eigenstates of this complete set of constants of motion, leads to the generalized Hankel transforms, used in a preceding article for the determination of the time dependent states for scattering and disintegration. In a representation in which the constants of motion are diagonal, the operator representing the relative kinetic energy of the two initial particles will be nondiagonal, particularly in the neighborhood of the resonance energy. The unitary matrix connecting any initial Fock state with the corresponding state at time $t$ will be obtained, and with its help the time-dependent behavior of the Heisenberg dynamical variables of our dynamical system can be derived.

Physical Review•Nov 1, 1951•14 citations

A stationary formulation of time-dependent problems in quantum mechanics

Peter Pfeifer, Raphael David Levine

A unifying framework is presented which treats time-dependent problems in nonrelativistic quantum mechanics on equal footing with stationary ones. This is accomplished by elevating time t to the role of a dynamical variable and considering evolution in an extended space with respect to a progress variable τ. Among the new objects in the extended space are the time operator as well as operators, such as energy, that are not diagonal in time. The latter bear, e.g., on the question of ‘‘quantum chaos.’’ Only τ-stationary states in the extended space are necessary to recover all of the Schrödinger time evolution. In particular, time-dependent constants of the motion (and these include time-evolved density matrices) elevate to stationary constants of the τ motion. Time-dependent problems, which may also involve time-dependent Hamiltonians, can then be solved by stationary-state methods. Special attention is given to applications based on time-dependent constants of the motion and to maximum-entropy states subject to time-dependent constraints. Two examples (rank-one perturbation, damped harmonic oscillator) illustrate some of these ideas. Scattering theory in the extended space is discussed in two appendices. It furnishes a picture of Hamiltonian τ evolution which promises to embrace also irreversible dynamics.

The Journal of Chemical Physics•Dec 1, 1983•50 citations

Asymptotic behavior in phase-space scattering

Juan Gonzalo Muga, Robert F. Snider

The phase-space formulation of quantum scattering is discussed and the asymptotic quantities describing potential scattering in one and three dimensions (transmittances, cross sections) are analyzed. It is found that under certain conditions the transmittance in time-dependent scattering only depends on the momentum distribution of the incident packet, being independent of its particular shape in coordinate space. For three-dimensional scattering, generalized cross sections are defined for momentum coherences and a physical interpretation of the generalized optical theorem is proposed.

Physical Review A•Mar 1, 1992•13 citations

Quantum Scattering Theory

Harish Parthasarathy

After a fairly rigorous mathematical formulation of quantum mechanics based on transformation theory by Dirac in which he cast Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics on a common operator theoretic setting and successfully explained their equivalence, it was John Von Neumann in monumental treatise “Mathematical foundations of quantum mechanics” who proved the spectral theorem for all self adjoint operators in an infinite dimensional Hilbert space and thereby cast the masterly work of Dirac “The principles of quantum mechanics“ in a way that even pure mathematicians were able to understand. Dirac in his book had also indicated the work of Jordan and Heisenberg on quantum scattering theory but without the modern method of wave operators. Lippmann and Schwinger gave a more rigorous discussion of scattering theory via the introduction of free particle states and scattered states. Actually, free particle states are not square integrable and hence cannot strictly be termed as states. However by the use of wave operators the equations of Lippmann and Schwinger could be made rigorous using the setting of the continuous spectrum of a self-adjoint opearator introduced by Von Neumann, Crucial to this development is the notion of the domain of definition of an unbounded operator. This leads to the fact that the wave operators that appear in scattering theory are not defined on all vectors but usually on those vectors that belong to the continuous spectrum of the free particle Hamiltonian. Today scattering theory is a well developed subject and the interested reader can get a good grasp of this subject from the books 1 Tosio Kato, “Perturbation theory for linear operators”, Springer. 2 Werner O. Amrein, “Hilbert space methods in quantum mechanics”, CRC press, 2012.

CRC Press eBooks•Sep 14, 2022•7 citations

"Quantum Mechanics in Mathematics, Chemistry, and Physics"

Karl E. Gustafson, William P. Reinhardt

Total Cross Sections in Non-Relativistic Scattering Theory.- Classical-Quantum Correspondence in Non-Linear Systems.- Real Axis Asymptotics and Estimates of Hamiltonian Resolvent Kernels.- Statistical Inference in Quantum Mechanics.- Schrodinger Operators with Oscillating Potentials.- Time Delay and Resonance in Simple Scattering.- A Study of the Helmholtz Operator.- Invariant Manifolds and Bifurcations in the Taylor Problem.- On the Dimension of a Finite Difference Approximation to Divergence-Free Vectors.- Intramolecular Dynamics in the Quasiperiodic and Stochastic Regimes.- Problems in the Semiclassical Quantization of Integrable and Nonintegrable Classical Dynamical Systems.- Nonselfadjoint Operators in Diffraction and Scattering.- One-Dimensional Crystals in an External Field.- Dynamics of Forced Coupled Oscillators: Classical Phenomenology of Infrared Multiphoton Absorption.- Irreversibility and Stochasticity of Chemical Processes.- On the Representations of the Local Current Algebra and the Group of Diffeomorphisms (I).- Regular and Chaotic Regimes in Quantum Mechanics.- Barrier Penetration and Exponential Decay in the Stark Effect.- A Nonlinear Schrodinger Equation Yielding the Shape of Molecules by Spontaneous Symmetry Breaking.- Classical Limit of the Number of Quantum States.- Formal Integrals for a Nonintegrable Dynamical System: Preliminary Report.- On a Generalized Weyl-Von Neumann Converse Theorem.- Scattering Theory in Many-Body Quantum Systems. Analyticity of the Scattered Matrix.- Existence, Uniqueness, Stability and Calculation of the Stationary Regimes in Some Nonlinear Systems.- Weighted Trigonometric Approximations in L2(Rn).- The Phenomenon of Magnetic Pairing and Exactly Soluble Models of Magnetic Resonances.- Dyons Solutions to Yang-Mills Equations.- Asymptotic Resonance Properties of the Finite-Dimensional Friedrichs Model.- Self-Consistent Field Methods for Metastable Electronic States: A Promising Extension of the Complex Coordinate Technique.- Timestep Control for the Numerical Solutions of Initial-Boundary-Value Problems.- Quantum Mechanical Angular Distributions and Group Representations on Banach Spaces.- Tensor Product Decomposition of Holomorphically Induced Representations and Clebsch-Gordan Coefficients.- Orthogonal Laurent Polynomials and Gaussian Quadrature.- Measurement Theory with Instruments Treated Partially Quantum Mechanically.- Transition Maps and Locality.- The Feynman-Kac Formula for Boson Wiener Processes.- On Nonunitary Equivalence Between Unitary Group of Dynamics and Contraction Semigroups of Markov Processes.

•Nov 1, 2011•50 citations

Analytic continuation of quantum systems and their temporal evolution

E. C. G. Sudarshan, Charles B. Chiu

The generalized vector space of quantum states is used to study the correspondence between the physical state space scrH and its continuation scrG. Consider the integral representation defined by the scalar product between an arbitrary vector in the dense subset of analytic vectors in scrH and its dual vector, where the integration is along the real axis. Keeping the scalar product fixed, the analytic vectors may be continued through the deformation of the integration contour. The deformed contour defines the generalized spectrum of the operator in the continued theory, which typically consists of a deformed contour in the fourth quadrant and the exposed singularities, if any, between the real axis and the deformed contour. Several models are studied with special attention to the unfolding of the generalized spectrum. The two-body models studied are the Lee model in the lowest sector and the Yamaguchi potential model, where the exposed singularities, if present, are simple poles. The three-body model studied is the cascade model, where the exposed singularities may be poles and the branch cuts associated with the quasi-two-body states. We demonstrate that the generalized spectrum obtained leads to the correct extended unitarity relation for the scattering amplitudes. The possibility of having mismatches between poles in the S matrix and the discrete states in the Hamiltonian, which exists in the scrH space, obtains also in the generalized scrG space. Finally, two distinct views on what constitutes an unstable particle are contrasted. One view is to identify it as a physical state of the system which ceases to exist as a discrete eigenstate in scrH. Here the survival amplitude of the unstable particle cannot be ever strictly exponential in time. The other view is to identify the unstable particle as a discrete state in the generalized space. It has a pure exponential time dependence. So the corresponding time evolution is realized by a semigroup. While the latter approach appears to be elegant, it is obtained at the expense of giving up the very starting premise of the lower boundedness of the energy spectrum and therefore we consider it to be the less desirable choice.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Mar 15, 1993•18 citations
Quantum mechanics gives us information about the spectra of dynamical variables and transition rates including scattering cross sections. They can be exhibited as spectral information in analytically continued spaces and their duals. Quantum mechanics formulated in these generalized spaces is used to study scattering and time evolution. It is shown that the usual asymptotic condition is inadequate to deal with the scattering of composite or unstable particles. Scattering theory needs an amendment when the interacting system is not isospectral with the free Hamiltonian; the amendment is formulated. Perturbation theory in generalized spaces is developed and used to study the deletion and augmentation of the spectrum of the Hamiltonian. A complete set of algebraically independent constants for an interacting system is obtained. The question of the breaking of time symmetry is discussed.
In the present paper we propose to develop a quantum-mechanical scheme in Fock space that would describe interactions that take place through the formation of a compound particle. The discussion will be restricted to a dynamical system representing a single-level scattering process. The state of this dynamical system can be found in two stages: initial particles and compound nucleus. The dynamical behavior of this state will be given in terms of a hamiltonian (that has no classical analog) which will be derived from the boundary conditions satisfied by the state. The representation of the dynamical variables, associated with the two stages of the state, will be discussed, and a complete set of constants of motion of our dynamical system will be given. The expansion of an arbitarary Fock state in terms of the simultaneous eigenstates of this complete set of constants of motion, leads to the generalized Hankel transforms, used in a preceding article for the determination of the time dependent states for scattering and disintegration. In a representation in which the constants of motion are diagonal, the operator representing the relative kinetic energy of the two initial particles will be nondiagonal, particularly in the neighborhood of the resonance energy. The unitary matrix connecting any initial Fock state with the corresponding state at time ttt will be obtained, and with its help the time-dependent behavior of the Heisenberg dynamical variables of our dynamical system can be derived.
A unifying framework is presented which treats time-dependent problems in nonrelativistic quantum mechanics on equal footing with stationary ones. This is accomplished by elevating time t to the role of a dynamical variable and considering evolution in an extended space with respect to a progress variable τ. Among the new objects in the extended space are the time operator as well as operators, such as energy, that are not diagonal in time. The latter bear, e.g., on the question of ‘‘quantum chaos.’’ Only τ-stationary states in the extended space are necessary to recover all of the Schrödinger time evolution. In particular, time-dependent constants of the motion (and these include time-evolved density matrices) elevate to stationary constants of the τ motion. Time-dependent problems, which may also involve time-dependent Hamiltonians, can then be solved by stationary-state methods. Special attention is given to applications based on time-dependent constants of the motion and to maximum-entropy states subject to time-dependent constraints. Two examples (rank-one perturbation, damped harmonic oscillator) illustrate some of these ideas. Scattering theory in the extended space is discussed in two appendices. It furnishes a picture of Hamiltonian τ evolution which promises to embrace also irreversible dynamics.
The phase-space formulation of quantum scattering is discussed and the asymptotic quantities describing potential scattering in one and three dimensions (transmittances, cross sections) are analyzed. It is found that under certain conditions the transmittance in time-dependent scattering only depends on the momentum distribution of the incident packet, being independent of its particular shape in coordinate space. For three-dimensional scattering, generalized cross sections are defined for momentum coherences and a physical interpretation of the generalized optical theorem is proposed.
After a fairly rigorous mathematical formulation of quantum mechanics based on transformation theory by Dirac in which he cast Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics on a common operator theoretic setting and successfully explained their equivalence, it was John Von Neumann in monumental treatise “Mathematical foundations of quantum mechanics” who proved the spectral theorem for all self adjoint operators in an infinite dimensional Hilbert space and thereby cast the masterly work of Dirac “The principles of quantum mechanics“ in a way that even pure mathematicians were able to understand. Dirac in his book had also indicated the work of Jordan and Heisenberg on quantum scattering theory but without the modern method of wave operators. Lippmann and Schwinger gave a more rigorous discussion of scattering theory via the introduction of free particle states and scattered states. Actually, free particle states are not square integrable and hence cannot strictly be termed as states. However by the use of wave operators the equations of Lippmann and Schwinger could be made rigorous using the setting of the continuous spectrum of a self-adjoint opearator introduced by Von Neumann, Crucial to this development is the notion of the domain of definition of an unbounded operator. This leads to the fact that the wave operators that appear in scattering theory are not defined on all vectors but usually on those vectors that belong to the continuous spectrum of the free particle Hamiltonian. Today scattering theory is a well developed subject and the interested reader can get a good grasp of this subject from the books 1 Tosio Kato, “Perturbation theory for linear operators”, Springer. 2 Werner O. Amrein, “Hilbert space methods in quantum mechanics”, CRC press, 2012.
Total Cross Sections in Non-Relativistic Scattering Theory.- Classical-Quantum Correspondence in Non-Linear Systems.- Real Axis Asymptotics and Estimates of Hamiltonian Resolvent Kernels.- Statistical Inference in Quantum Mechanics.- Schrodinger Operators with Oscillating Potentials.- Time Delay and Resonance in Simple Scattering.- A Study of the Helmholtz Operator.- Invariant Manifolds and Bifurcations in the Taylor Problem.- On the Dimension of a Finite Difference Approximation to Divergence-Free Vectors.- Intramolecular Dynamics in the Quasiperiodic and Stochastic Regimes.- Problems in the Semiclassical Quantization of Integrable and Nonintegrable Classical Dynamical Systems.- Nonselfadjoint Operators in Diffraction and Scattering.- One-Dimensional Crystals in an External Field.- Dynamics of Forced Coupled Oscillators: Classical Phenomenology of Infrared Multiphoton Absorption.- Irreversibility and Stochasticity of Chemical Processes.- On the Representations of the Local Current Algebra and the Group of Diffeomorphisms (I).- Regular and Chaotic Regimes in Quantum Mechanics.- Barrier Penetration and Exponential Decay in the Stark Effect.- A Nonlinear Schrodinger Equation Yielding the Shape of Molecules by Spontaneous Symmetry Breaking.- Classical Limit of the Number of Quantum States.- Formal Integrals for a Nonintegrable Dynamical System: Preliminary Report.- On a Generalized Weyl-Von Neumann Converse Theorem.- Scattering Theory in Many-Body Quantum Systems. Analyticity of the Scattered Matrix.- Existence, Uniqueness, Stability and Calculation of the Stationary Regimes in Some Nonlinear Systems.- Weighted Trigonometric Approximations in L2(Rn).- The Phenomenon of Magnetic Pairing and Exactly Soluble Models of Magnetic Resonances.- Dyons Solutions to Yang-Mills Equations.- Asymptotic Resonance Properties of the Finite-Dimensional Friedrichs Model.- Self-Consistent Field Methods for Metastable Electronic States: A Promising Extension of the Complex Coordinate Technique.- Timestep Control for the Numerical Solutions of Initial-Boundary-Value Problems.- Quantum Mechanical Angular Distributions and Group Representations on Banach Spaces.- Tensor Product Decomposition of Holomorphically Induced Representations and Clebsch-Gordan Coefficients.- Orthogonal Laurent Polynomials and Gaussian Quadrature.- Measurement Theory with Instruments Treated Partially Quantum Mechanically.- Transition Maps and Locality.- The Feynman-Kac Formula for Boson Wiener Processes.- On Nonunitary Equivalence Between Unitary Group of Dynamics and Contraction Semigroups of Markov Processes.
The generalized vector space of quantum states is used to study the correspondence between the physical state space scrH and its continuation scrG. Consider the integral representation defined by the scalar product between an arbitrary vector in the dense subset of analytic vectors in scrH and its dual vector, where the integration is along the real axis. Keeping the scalar product fixed, the analytic vectors may be continued through the deformation of the integration contour. The deformed contour defines the generalized spectrum of the operator in the continued theory, which typically consists of a deformed contour in the fourth quadrant and the exposed singularities, if any, between the real axis and the deformed contour. Several models are studied with special attention to the unfolding of the generalized spectrum. The two-body models studied are the Lee model in the lowest sector and the Yamaguchi potential model, where the exposed singularities, if present, are simple poles. The three-body model studied is the cascade model, where the exposed singularities may be poles and the branch cuts associated with the quasi-two-body states. We demonstrate that the generalized spectrum obtained leads to the correct extended unitarity relation for the scattering amplitudes. The possibility of having mismatches between poles in the S matrix and the discrete states in the Hamiltonian, which exists in the scrH space, obtains also in the generalized scrG space. Finally, two distinct views on what constitutes an unstable particle are contrasted. One view is to identify it as a physical state of the system which ceases to exist as a discrete eigenstate in scrH. Here the survival amplitude of the unstable particle cannot be ever strictly exponential in time. The other view is to identify the unstable particle as a discrete state in the generalized space. It has a pure exponential time dependence. So the corresponding time evolution is realized by a semigroup. While the latter approach appears to be elegant, it is obtained at the expense of giving up the very starting premise of the lower boundedness of the energy spectrum and therefore we consider it to be the less desirable choice.