Matrix theory of small oscillations

Chavda, L. K.

doi:10.1119/1.11243

Cited by 5 publications

(9 citation statements)

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“…The parameters a and b that characterize the general linear coordinate transformation (10) and (11) can take, in principle, any value in the interval (-¥ +¥ , ), except for those with a=1/b, which make coordinates z 1 and z 2 linearly dependent. There are, however, pairs of (a, b) values that provide essentially the same set of coordinates z 1 and z 2 .…”

Section: Linear Coordinate Transformation and Hamiltonian Operadormentioning

confidence: 99%

“…By writing the transformation equations (10) and (11) as a function of the angular parameters α and β we get…”

Section: Linear Coordinate Transformation and Hamiltonian Operadormentioning

confidence: 99%

“…The harmonic oscillator model plays a fundamental role in describing the structure of matter and its interactions [1][2][3][4][5][6]. In coupled vibrational systems, the model provides the normal coordinates, which eliminate the linear couplings and thus allow the description of the system by the corresponding uncoupled normal modes of vibration [3,4,[7][8][9][10]. Normal coordinates are therefore of crucial importance in the characterisation of coupled vibrational systems and become the natural starting point for the development of more elaborated and accurate representations of them [11][12][13][14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

“…The usual way to construct normal coordinates is to perform a linear transformation of the original coordinates of the coupled system and determine the transformation coefficients by removing the linear couplings [3,9,10]. Linear transformations can, however, go further in this context.…”

Section: Introductionmentioning

confidence: 99%

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Quantum description of linearly coupled harmonic oscillator systems using oblique coordinates

Zúñiga

Bastida

Requena

2019

J. Phys. B: At. Mol. Opt. Phys.

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In this article we extend our previous quantum-mechanical treatment of the system of identical harmonic oscillators linearly coupled in the kinetic and potential energies using oblique coordinates (Zúñiga et al 2019 J. Phys. B: At. Mol. Opt. Phys. 52 055101) to the general system of coupled non-identical harmonic oscillators. Oblique coordinates are obtained by making non-orthogonal rotations of the original coordinates that convert the matrix representation of the quadratic Hamiltonian operator into a block diagonal matrix. Accordingly, we derive the analytical formula for the oblique rotation angles, and obtain the expressions, also analytical, for the energy levels and eigenfunctions of the system in terms of the oblique parameters. We also show that oblique coordinates are in fact dependent on one of the rotational angles whose value can be freely chosen, so they form, in fact, a continuous set of coordinates that are especially flexible in dealing with more complex vibrational systems. To illustrate this, we make a numerical application to a system of kinetically coupled Barbanis oscillators, which clearly shows the advantages of using oblique coordinates to determine the energy levels and wave functions of the system variationally.

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Section: Linear Coordinate Transformation and Hamiltonian Operadormentioning

confidence: 99%

“…By writing the transformation equations (10) and (11) as a function of the angular parameters α and β we get…”

Section: Linear Coordinate Transformation and Hamiltonian Operadormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantum description of linearly coupled harmonic oscillator systems using oblique coordinates

Zúñiga

Bastida

Requena

2019

J. Phys. B: At. Mol. Opt. Phys.

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“…One of the keys to the usefulness of this type of system lies in the fact that when couplings are bilinear, i.e. quadratic, it is relatively easy to obtain the exact solutions of both classical and quantum equations of motion numerically by the corresponding transformation to normal coordinates [4,6,[11][12][13][14][15], a transformation that in some cases can be done analytically [3-5, 11, 16, 17].…”

Section: Introductionmentioning

confidence: 99%

Quantum solutions of identical linearly coupled harmonic oscillators using oblique coordinates

Zúñiga¹,

Bastida²,

Requena³

2019

J. Phys. B: At. Mol. Opt. Phys.

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In this article we develop the solutions of the time-independent Schrödinger equation analytically for a system of two identical harmonic oscillators linearly coupled in the kinetic and potential energies, using oblique coordinates. These coordinates are constructed by making a non-orthogonal rotation of the original coordinates that allows us to write the matrix representation of the Hamiltonian operator in a block-diagonal form characterized by the polyadic quantum number n = n1 + n2. The expression for the oblique rotation angle is first derived as a function of the kinetic and potential coupling parameters. Then the expression for the exact energy levels of the system is obtained by making an orthogonal rotation of the oblique coordinates, and the structure of the spectrum is discussed. Finally, the eigenvectors of the Hamiltonian polyadic blocks are identified with the discrete orthogonal polynomials of Krawtchouk, which provides an analytical expression for the coefficients of the linear combinations of the wave functions in oblique coordinates.

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Method of normal coordinates in the formulation of a system with dissipation: the harmonic oscillator

Mshelia

1995

Nuov Cim A

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Matrix theory of small oscillations

Abstract: We present a complete matrix formulation of the theory of small oscillations. Simple analytic solutions involving matrix functions are found which clearly exhibit the transients, the damping factors, the Breit-Wigner form for resonances, etc.

Cited by 5 publications

References 0 publications

Quantum description of linearly coupled harmonic oscillator systems using oblique coordinates

Quantum description of linearly coupled harmonic oscillator systems using oblique coordinates

Quantum solutions of identical linearly coupled harmonic oscillators using oblique coordinates

Method of normal coordinates in the formulation of a system with dissipation: the harmonic oscillator

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