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Physical Review A

Motion of the wave-function zeros in spin-boson systems

Demosthenes Ellinas, Vassilios I. Kovanis

Published May 1, 1995• 6 citations
Publisher

Abstract

In the analytic Bargmann representation associated with the harmonic oscillator and spin coherent states, the wave functions considered as consisting of entire complex functions can be factorized in terms of their zeros in a unique way. The Schr\"odinger equation of motion for the wave function is turned to a system of equations for the zeros of the wave function. The motion of these zeros as a nonlinear flow of points is studied and interpreted for linear and nonlinear bosonic and spin Hamiltonians. Attention is given to the study of the zeros of the Jaynes-Cummings model and to its finite analog. Numerical solutions are derived and dicussed.

DOI
10.1103/physreva.51.4230

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In the analytic Bargmann representation associated with the harmonic oscillator and spin coherent states, the wave functions considered as consisting of entire complex functions can be factorized in terms of their zeros in a unique way. The Schr"odinger equation of motion for the wave function is turned to a system of equations for the zeros of the wave function. The motion of these zeros as a nonlinear flow of points is studied and interpreted for linear and nonlinear bosonic and spin Hamiltonians. Attention is given to the study of the zeros of the Jaynes-Cummings model and to its finite analog. Numerical solutions are derived and dicussed.
In the analytic-Bargmann representation associated with the harmonic oscillator and spin coherent states, the wavefunction as entire complex functions can be factorized in terms of their zeros in a unique way. The Schrödinger equation of motion for the wavefunction is turned to a system of equations for its zeros. The motion of these zeros as a non-linear flow of points is studied and interpreted for linear and non-linear bosonic and spin Hamiltonians. Attention is given to the study of the zeros of the Jaynes-Cummings model and to its finite analoque. Numerical solutions are derived and discussed.
Replaces the Schrodinger equation for the time propagation of states of a quantized 2D spherical phase space by the dynamics of a system of N particles lying in phase space. This is done through factorization formulae of an analytic function theory arising in a coherent-state representation, the 'particles' being the zeroes of the quantum state. For linear Hamiltonians, like a spin in a uniform magnetic field, the motion of the particles is classical. However, nonlinear terms induce interactions between the particles. Their time propagation is studied and it is shown that, contrary to integrable systems, for chaotic maps they tend to fill, as their classical counterpart, the whole phase space in a uniform way.
We consider a spin coherent states description of a general quantum spin system. It is shown that it is possible to use the spin-1/2 representation to study the general spin-J case. We identify the 1/2 spinor components as the homogeneous coordinates of the projective space associated to the complex variable that labels the coherent states and establish a relation between the two-component spinor and the bosonic Schwinger representation of a spin operator. We rewrite the equations of motion, obtained from the path integral for the evolution operator or partition function, in terms of the 1/2 spinor and define the effective Hamiltonian of its evolution.
The paper contains an investigation of zeros Of Bargmann analytic representation. A brief introduction to Harmonic oscillator formalism is given. The Bargmann analytic representation has been studied. The zeros of Bargmann analytic function are considered. The Q or Husimi functions are introduced. The The Bargmann functions and the Husimi functions have the same zeros. The Bargmann functions f(z) have exactly q zeros. The evolution time of the zeros μn are discussed. Various examples have been given.
Various Euclidean, hyperbolic and elliptic analytic representations are introduced and relations among them are discussed. The Bargmann analytic representation in the complex plane is considered and its relation to other phase-space methods for the harmonic oscillator is reviewed. The general theory that relates the growth of analytic functions with the density of their zeros is applied to Bargmann functions and it leads to theorems on the completeness of sequences of Glauber coherent states. Two hyperbolic analytic representations in the unit disc, based on SU (1, 1) coherent states and also on phase states are introduced. A third analytic representation in the complex plane based on Barut–Girardello states is also considered and transformations which relate it to the other ones are studied. In the case of systems with finite-dimensional Hilbert space, an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced. The Berezin formalism in the Euclidean, hyperbolic and elliptic cases is discussed. Contour analytic representations in these three cases are also presented.
We consider the nature of the wave function using the example of a harmonic oscillator. We show that the eigenfunctions ψn=znψ_n{=}z^nψn​=zn of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation with z∈Cz\in\mathbb Cz∈C are the coordinates of a classical oscillator with energy En=ℏωnE_n=\hbarωnEn​=ℏωn, n=0,1,2,... n=0,1,2,...\,n=0,1,2,.... They are defined on conical spaces C/Zn{\mathbb C}/{\mathbb Z}_nC/Zn​ with cone angles 2π/n2π/n2π/n, which are embedded as subspaces in the phase space C\mathbb CC of the classical oscillator. Here Zn{\mathbb Z}_nZn​ is the finite cyclic group of rotations of the space C\mathbb CC by an angle 2π/n2π/n2π/n. The superposition ψ=∑ncnψnψ=\sum_n c_nψ_nψ=∑n​cn​ψn​ of the eigenfunctions ψnψ_nψn​ arises only with incomplete knowledge of the initial data for solving the Schrödinger equation, when the conditions of invariance with respect to the discrete groups Zn{\mathbb Z}_nZn​ are not imposed and the general solution takes into account all possible initial data parametrized by the numbers n∈Nn\in\mathbb Nn∈N.