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Journal of Physics A Mathematical and General

Solution of the eigenvalue problem of a N-level system coupled to a bosonic degree of freedom without using RWA

N Klenner, Jasmin Weis

Published Apr 1, 1987• 3 citations
Publisher

Abstract

The eigenvalue problem of a N-level system coupled to a bosonic degree of freedom is solved without using RWA. For that purpose, the bosonic degree of freedom is transformed to Bargmann's Hilbert space of analytical functions. In this representation the Schrodinger equation is a system of N coupled linear differential equations of first order. Using a discrete symmetry, these equations are simplified by a suitable transformation of the independent variable. Starting from the simplified equations, the authors develop a method to solve the eigenvalue problem of the N-level system. In addition, they present a simple approximate treatment and compare it with the exact results. The approximation turns out to be quite good up to outer level resonance and can be used to explain the differently structured regions in the energy spectra.

DOI
10.1088/0305-4470/20/5/025

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The eigenvalue problem of a N-level system coupled to a bosonic degree of freedom is solved without using RWA. For that purpose, the bosonic degree of freedom is transformed to Bargmann's Hilbert space of analytical functions. In this representation the Schrodinger equation is a system of N coupled linear differential equations of first order. Using a discrete symmetry, these equations are simplified by a suitable transformation of the independent variable. Starting from the simplified equations, the authors develop a method to solve the eigenvalue problem of the N-level system. In addition, they present a simple approximate treatment and compare it with the exact results. The approximation turns out to be quite good up to outer level resonance and can be used to explain the differently structured regions in the energy spectra.
The Hamiltonian of a two-level system coupled to a boson mode is transformed to Bargmann's Hilbert space of analytical functions. The quantum dynamics is calculated with a suitable choice of a complete set of orthonormal functions in Bargmann space. A semiclassical approximation which is formulated as a dynamical Hamiltonian system is compared with the quantum dynamics. For the parameters under consideration, semiclassical dynamics is not at all appropriate to approximate the quantum dynamics. These parameters are suited to view the system as a two-site model of small-polaron dynamics.
We demonstrate that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the bound-state problem in a single well of one degree of freedom. While the WKB expansion formulas are basically the usual ones, in this representation they describe approximations that are uniform and nonsingular in the classically allowed region of phase space because no turning points appear there. The quantization of energy levels relies on a complex contour integral that tests the eigenfunction for analyticity. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a self-consistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions. All in all, the Bargmann representation appears to combine the advantages of a linear description and of a phase-space representation of the quantum state vectors.
Recently Zhang (2013 J. Phys. A: Math. Theor. 46 455302) proposed an analytical approach to solve the time-independent Schrödinger equation for the single-mode and two-mode squeezed harmonic oscillators in the Bargmann space of entire functions. In this comment we show that the eigenfunctions of these two systems exist in closed form and are expressed in terms of the Hermite polynomials. Moreover, since both oscillators exhibit the SU(1,1) dynamical symmetry, the eigenvalue problem can be tackled in a unified manner. In the Hilbert space of analytic functions of a complex variable in the unit disc, the energy eigenvalue equations involve first-order ordinary differential equations only, so we can easily solve these equations to obtain simple closed-form solutions.
This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.