• P M Mathews

      Articles written in Pramana – Journal of Physics

    • Covariant fields: Poincaré group representations and metric structure in the space of quantum states

      P M Mathews

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      The representations of the Poincaré group realized over the space of covariant fields transforming according to any irreducible representationD(m,n) of the Lorentz group are constructed explicitly with reference to a helicity basis. The representation is indecomposable in the massless case. The form of this representation together with the invariance of two-point Wightman functions of the field (which follows from a weak set of axioms) determines the metric structure in the space of quantum states of the field. This structure is explicitly determined for generalD(m,n). Certain particular cases (especially the symmetric traceless tensor field) are discussed in detail. Finally we consider the representation pertaining to massive fields, and examine the passage to the limit of vanishing mass. We present a limiting procedure which leads from the unitary representation of the massive field to the indecomposable non-unitary representation of the massless field.

    • Residue-squaring iterative diagonalization method for perturbation problems

      P M Mathews

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      We present a new method for the evaluation of the change in eigenvalues due to a perturbation of strength λ. It is a fast converging iterative method which, at thenth step, gives results accurate to order (2n+1−1) in λ. Unlike the Rayleigh-Schrödinger perturbation theory in quantum mechanics, which becomes prohibitively cumbersome when carried to higher orders, the present method involves a routine which remains stralghtforward at all stages.

    • Residue-squaring diagonalisation method and the anharmonic oscillator

      P M Mathews T R Govindarajan

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      A recently-formulated residue-squaring method for perturbation problems is subjected to an exacting test in its application to the problem of diagonalising the Hamiltonian of the nonlinear oscillator with quartic anharmonicity. Unlike other methods, this new iterative diagonalisation method enables several eigenvalues to be calculated simultaneously with little more labour than for a single eigenvalue. Values obtained for the four lowest even-parity levels of the anharmonic oscillator from just two or three iterations are shown to agree well with earlier accurate calculations. An approximate analytical formula for the energy levels is also presented.

    • On the problem of constraints in minimally coupled relativistic wave equations for particles of unique mass

      M Seetharaman T R Govindarajan P M Mathews

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      We study the problem of a possible change in the number of constraints in linear relativistic wave equations (-μμ+m)ψ=0 for particles of unique mass, on introduction of minimal coupling to an external electromagnetic field. Complementing our earlier work in which we obtained conditions for non-loss of constraints in equations characterised by the minimalβ-algebraβ05 =β03 we derive here the conditions for such theories not to generate more constraints than in the free case. The results are illustrated by considering specific equations and a fallacy in certain conclusions of Kobayashi and Shamaly on this problem is pointed out.

    • On the energy spectra of one-dimensional anharmonic oscillators

      P M Mathews M Seetharaman Sekhar Raghavan V T A Bhargava

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      In this paper we present explicit and simple analytical formulae for the energy eigenvaluesEn (λ) of one-dimensional anharmonic oscillators characterized by the potentials 1/22x2x withα=2, 3 and 4. A simple intuitive criterion supplemented by the requirement of correct asymptotic behaviour, has been employed in arriving at the formulae. Our energy values over a wide range ofn andλ are in good agreement with the numerical values computed by earlier workers through very elaborate techniques. To our knowledge this is the first time that formulae of such wide validity have been given. The results for pure power oscillators are trivially obtained by going over to theω→0 limit. Approximate analytic expressions for the low order even moments ofx are also given.

    • Determination of eigenvalues of real symmetric para-p diagonal matrices

      V T A Bhargava P M Mathews M Seetharaman

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      A method is presented for an accurate numerical determination of eigenvalues of real symmetric para-p diagonal matrices. The method takes advantage of the band structure to break up the matrix intop ×p blocks and performing algebraic operations including inversions on these blocks only, no matter what the size of the matrix is. The eigenvalues are determined independently one at a time. Thus any error in the determination of one eigenvalue does not affect the other eigenvalues. The method is ideally suited for the Schrödinger eigen alue problem of the anharmonic potentials, which is taken up in the following paper.

    • Anharmonic oscillators in higher dimension: Accurate energy eigenvalues and matrix elements

      V T A Bhargava P M Mathews M Seetharaman

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      Energy eigenvalues and matrix elements of various anharmonic oscillators are determined to a high accuracy by applying a method for determining the eigenvalues and eigenvectors of real symmetric para-p diagonal matrices (described in the preceding paper). Our results for the 2- and 3-dimensional oscillators are new and complement similar accurate results for the one dimensional oscillators available in the literature.

    • Anharmonic oscillators in higher dimension: Accurate energy eigenvalues and matrix elements

      V T A Bhargava P M Mathews M Seetharaman

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