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Journal of Physics B Atomic Molecular and Optical Physics

Quantum solutions of identical linearly coupled harmonic oscillators using oblique coordinates

José Zúñiga, Adolfo Bastida, Alberto Requena

Published Jan 3, 2019• 5 citations
Publisher

Abstract

Abstract 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 = n 1 + n 2 . 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.

DOI
10.1088/1361-6455/aafbe1

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The normal frequencies and its corresponding normal coordinates and conjugate momentums of the n-module coupled harmonic oscillators quantum system are derived by using the invariant eigen-operator method.The Hamiltonian of the system is decoupled and then the visible analytic solution of normal frequencies in the system is obtained.The analytic solution of wave function of the system in the coordinate representation is deduced.Through the discussion of the coupling coefficients of different situations,we have concluded that the coupled harmonic oscillator systems with the n-module momentum or coordinates are just special cases of the system under study.
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Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product