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MANY LEVEL SCATTERING IN FOCK SPACE I. II. III

F.M.M. Nicolau

Published Jan 1, 1954• 0 citations
OpenAlex

Abstract

The time-dependent description of the many level scattering is discussed. The interaction is assumed to take place through the formation of a compound nucleus, and the description is given in an appropriate Fock space. In particular the Hamiltonian of the problem is derived and the quantum mechanical representation of the potential is given this potential having no classical analog. (auth)

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Journal of Physics A Mathematical and General•May 10, 2005•1 citation
The time-dependent description of the many level scattering is discussed. The interaction is assumed to take place through the formation of a compound nucleus, and the description is given in an appropriate Fock space. In particular the Hamiltonian of the problem is derived and the quantum mechanical representation of the potential is given this potential having no classical analog. (auth)
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.
In the present paper the quantum mechanical representation in phase space introduced by Torres-Vega and Frederick (1993 J. Chem. Phys. 98 3103) has been successfully applied to the phase space description of the quantum potential scattering. 3D potential scattering is formulated from the view-points of the wave function and density operator in this new quantum phase-space representation. It is found that in the region where the potential acts predominantly, the scattered part of the wave function or density in quantum phase space depends closely not only on the potential, but also on a non-locality factor arising from the use of the eigenfunctions of the coordinate and momentum operators in the phase-space representation. Transition from the present formulation to the classical scattering is made.
Necessary conditions under which a classical description will give the correct quantum relaxation behavior are analyzed. Assuming a nonequilibrium preparation, it is shown that the long-time mean values of observables can be expressed in terms of the spectral density and state-specific level densities of the system. Any approximation that reproduces these quantities therefore yields the correct expectation values at long times. Apart from this rigorous condition, a weaker but more practical criterion is established, that is, to require that the total level density is well approximated in the energy range defined by the spectral density. Since the integral level density is directly proportional to the phase-space volume that is energetically accessible to the system, the latter condition means that an appropriate classical approximation should explore the same phase-space volume as the quantum description. In general, however, this is not the case. A well-known example is the unrestricted flow of zero-point energy in classical mechanics. To correct for this flaw of classical mechanics, quantum corrections are derived which result in a restriction of the classically accessible phase space. At the simplest level of the theory, these corrections are shown to correspond to the inclusion of only a fraction of the full zero-point energy into the classical calculation. Based on these considerations, a general strategy for the classical simulation of quantum relaxation dynamics is suggested. The method is (i) dynamically consistent in that it refers to the behavior of the ensemble rather than to the behavior of individual trajectories, (ii) systematic in that it provides (rigorous as well as minimal) criteria which can be checked in a practical calculation, and (iii) practical in that it retains the conceptional and computational simplicity of a standard quasiclassical calculation. Employing various model problems which allow for an analytical evaluation of the quantities of interest, the virtues and limitations of the approach are discussed.
Two schemes of the quantization of dynamical variables of dissipative systems are studied with an example of a two-level atom interacting with phonons scattered from an environment. Transformations of relevant Fock-space operators or rigged-Fock-space operators are proposed in order to diagonalize their equations of motion. It is shown that the damping of dispersionless phonons leads to the formation of a quasi-stationary state of the system—a polaron, which is found in both approaches. A time dependence of the polaron energy and the spectral lineshape of the two-level subsystem are evaluated. A transformation diagonalizing the equations of motion of the relevant classical problem (of a harmonic oscillator coupled to a field of damped waves) is found to be similar to the quantum case.