Quantum solutions of identical linearly coupled harmonic oscillators using oblique coordinates

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 = 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.

Quantum Dynamics of Oblique Vibrational States in the Hénon–Heiles System

“…In view of our recent work on the usefulness of oblique coordinates in two-dimensional systems, both in the time-independent − and time-dependent (present work) approaches, it is clear that the next step is to extend oblique coordinates to systems with a larger number of degrees of freedom such as the vibrational motions taking place in polyatomic molecules and in condensed phase systems. One of the advantages of oblique coordinates over normal coordinates is the greater number of parameters available in the former, which is useful to optimize them conveniently with greater flexibility.…”

“…The transformation from normal to local coordinates reintroduces therefore these couplings. , Generally, this transformation is linear and usually orthogonal, meaning that local coordinates are expressed as an orthogonal combination of normal coordinates. However, this linear combination is not the only one that can be used. − Our group has been working for a long time on the possibility of performing generalized nonorthogonal linear transformations to describe the vibrational stretching motions of triatomic molecules, − and we have recently focused on studying in detail this type of transformation that allows for the second-order Hamiltonian matrix of the system to be expressed in a block diagonal form, employing coupled oscillator models. − The resulting coordinates can be visualized as individual nonorthogonal rotations of the original normal coordinates, that is, oblique coordinates …”

Quantum description of linearly coupled harmonic oscillator systems using oblique coordinates

“…Our group has recently described in detail the quantum treatment of a system of two identical harmonic oscillators linearly coupled in the kinetic and potential energies, using oblique coordinates [25]. In the present work we extend our earlier treatment to the general system of non-identical linearly coupled harmonic oscillators.…”

“…( ) ¢ = k k . 9 12 12 It is now that we make the generalized linear coordinate transformation [24,25] ( where a and b are the transformation parameters and where the normalization factor 1/(1−ab) 1/2 guarantees the Jacobian is the unit. The Hamiltonian operator based on coordinates z 1 and z 2 is written then as and the reduced masses μ ij and force constants U 11 , U 22 and U 12 are given by ⎡…”

“…Quantum-mechanically, it is possible to construct them so that they make the harmonic matrix representation of the quadratic Hamiltonian operator have a block diagonal structure characterized by the polyadic quantum number n=n 1 +n 2 , thus providing the so-called oblique coordinates, which visualize the linear transformation as specific non-orthogonal rotations of the original coordinates describing the system [23][24][25].…”