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arXiv:nucl-th/0611014 · Physical Review Letters

Equations-of-Motion Approach to Quantum Mechanics: Application to a Model Phase Transition

Shen Yong Ho, George Rosensteel, DAVID J. ROWE

Published Feb 20, 2007• 4 citations
arXivHTMLPDF

Abstract

We present a generalized equations-of-motion method that efficiently calculates energy spectra and matrix elements for algebraic models. The method is applied to a five-dimensional quartic oscillator that exhibits a quantum phase transition between vibrational and rotational phases. For certain parameters, 10 x 10 matrices give better results than obtained by diagonalizing 1000 x 1000 matrices.

DOI
10.1103/physrevlett.98.080401

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We present a generalized equations-of-motion method that efficiently calculates energy spectra and matrix elements for algebraic models. The method is applied to a five-dimensional quartic oscillator that exhibits a quantum phase transition between vibrational and rotational phases. For certain parameters, 10 x 10 matrices give better results than obtained by diagonalizing 1000 x 1000 matrices.
An algebraic approach, within a framework using equations of motion, is applied to the study of vibrational collective motion in exactly soluble shell models with symmetry R(5) and R(5) \ifmmode\times\else\texttimes\fi{} R(5). The latter exhibits two phonon branches, permitting the study of the effect of the noncollective on the collective branch. The results of this method, including the splitting of the two-phonon degeneracy, agree satisfactorily with exact results for a range of parameters well into the transition region between vibrational and rotational motion.[NUCLEAR STRUCTURE Approximate many-body methods applied to a solvable model. Vibrational spectrum. Equations of motion method.]
Matrices of dynamical variables of the one-dimensional harmonic oscillator are calculated by a new semiclassical method formulated by More and Warren. The results are semiclassical matrices Xn, m, Pn, m, etc., which i) exactly obey selection rules for zero matrix-elements, ii) are very accurate, circa 1 %, for nonzero matrix-elements, iii) which diagonalize the Hamiltonian H, and iv) which exactly satisfy the Heisenberg commutation relations and equations of motion. These statements are true for matrix-elements Fn, m with m ≽ n. The results show that most of the quantum physics is contained in the extended semiclassical theory.
This chapter aims to develop an appropriate notation and to facilitate the subsequent discussion of the role played by the Lie algebras. It is unlikely that one manages to express a sufficiently realistic model only in terms of the operators that span a finite-dimensional Lie algebra, but it may be possible to simulate most of the relevant features of many physical phenomena by means of such simple models. The chapter discusses well-known approximate methods for the treatment of the time-evolution equation. If the Hamiltonian operator belongs to a finite-dimensional Lie algebra, then the differential equation for the time-evolution operator is easier to solve, and even the calculation of transition probabilities and expectation values of observables may be straightforward. The most relevant feature of this application of the Lie algebraic method is the occurrence of a matrix equation of finite dimension in contrast with the expansion in a basis set of state vectors, which typically leads to matrices of infinite dimension.
The algebraic structure of a truncated semiclassical Hamiltonian describing the translational–vibrational interaction has been determined and exploited to compute the S matrix for the system. The equations of motion have been integrated in a 4×4 faithful nonunitary representation of the group, and the final results lifted to the infinite dimensional unitary representation of the group. Computations of transition probabilities are facilitated by using powerful disentangling theorems to write the S matrix in convenient operator-ordered form. These methods are in quantitative agreement with results of Gazdy and Micha, and generalize easily to the multimode case. The structure of the Ehrenfest equations of motion for the operators of the algebra is identical to the structure of the dynamical algebra itself. The S matrix is also reconstructed from these equations of motion. Finally, a simple geometric picture is presented to describe the two extreme limits of the parametric oscillator problem involving only single quantum transitions or double quantum transitions.
Theory of the quantum quartic oscillator is developed with close attention to the energy cutoff one needs to impose on the system in order to approximate the smallest eigenvalues and corresponding eigenstates of its Hamiltonian by diagonalizing matrices of limited size. The matrices are obtained by evaluating matrix elements of the Hamiltonian between the associated harmonic-oscillator eigenstates and by correcting the computed matrices to compensate for their limited dimension, using the Wilsonian renormalization-group procedure. The cutoff dependence of the corrected matrices is found to be described by a spiral motion of a three-dimensional vector. This behavior is shown to result from a combination of a limit-cycle and a floating fixed-point behaviors, a distinct feature of the foundational quantum system that warrants further study. A brief discussion of the research directions concerning renormalization of polynomial interactions of degree higher than four, spontaneous symmetry breaking and coupling of more than one oscillator through the near neighbor couplings known in condensed matter and quantum field theory, is included.
The original version of matrix mechanics is little used in applications nowadays. In that early theory position, momentum and other quantities were represented by infinite matrices, whereas many modern applications of quantum mechanics represent such observables as operators. Infinite matrices are unwieldy to use and have some disturbing properties, e.g. they do not obey the associative rule. By far the most commonly appearing matrix in quantum mechanics is the matrix of the energy operator as set up in some finite basis of trial functions. The procedure has been used for perturbed oscillator problems; the recurrence relation approach is particularly simple when the matrix has only one non-zero element beyond the diagonal element in each row. It seems to the author that there are still many hybrid methods which need investigation.