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Journal of Mathematical Physics

Coupled harmonic oscillator systems: An elementary algebraic decoupling approach

Tova Feldmann, M. Cohen, B. L. Burrows

Published Sep 1, 2000• 4 citations
Publisher

Abstract

We present simple explicit coordinate transformations which serve to decouple the Schrödinger equation for a pair of (not necessarily identical) harmonic oscillators in the presence of bilinear perturbing potentials. We derive general conditions for the decoupling, and give some examples of physical interest. These include the much studied example with just a static perturbation, the parallel problem with a dynamic coupling term, and the classic example of an isotropic two-dimensional oscillator in a transverse magnetic field, first solved by Fock (1928) by separation of variables.

DOI
10.1063/1.1288248

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We present simple explicit coordinate transformations which serve to decouple the Schrödinger equation for a pair of (not necessarily identical) harmonic oscillators in the presence of bilinear perturbing potentials. We derive general conditions for the decoupling, and give some examples of physical interest. These include the much studied example with just a static perturbation, the parallel problem with a dynamic coupling term, and the classic example of an isotropic two-dimensional oscillator in a transverse magnetic field, first solved by Fock (1928) by separation of variables.
Abstract We present a novel explicit decoupling procedure that leads to exact solutions of the Schrödinger equation forNharmonic oscillators in the presence of general bilinear perturbing potentials. This generalizes previous work, which was restricted toorthogonaltransformations with conditions imposed on the perturbing terms. It is shown that the formalism is sufficiently general to embrace bothstationary‐stateandresonance‐stateproblems and, in particular, we include exact model calculations of a free electron interacting with a lattice and a discussion of an application to an ideal ion trap. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
It is one of the important problems to solve the Schrodinger equation for the coupled harmonic oscillators.Here the Heisenberg equations of motion for the coupled harmonic oscillators are written in the form of Schrodinger type equation,which can be decoupled by certain transformation.Further,the Hamiltonian can be diagonalized.For the anisotropic harmonic oscillator in a background magnetic field,the diagonalization form of the Hamiltonian,the energy eigenvalues and the eigenstates are derived.Such a process of derivation has clear physical picture and is easy to understand.
In many applications of quantum optics, nonlinear physics, molecular chemistry and biophysics, one can encounter models in which the coupled quantum harmonic oscillator provides an explanation for many physical phenomena and effects. In general, these are harmonic oscillators coupled via coordinates and momenta, which can be represented as H^=∑i=12p^i22mi+miωi22xi2+H^int, where the interaction of two oscillators H^int=ik1x1p^2+ik2x2p^1+k3x1x2−k4p^1p^2. Despite the importance of this system, there is currently no general solution to the Schrödinger equation that takes into account arbitrary initial states of the oscillators. Here, this problem is solved in analytical form, and it is shown that the probability of finding the system in any states and quantum entanglement depends only on one coefficient R∈(0,1) for the initial factorizable Fock states of the oscillator and depends on two parameters R∈(0,1) and ϕ for arbitrary initial states. These two parameters R∈(0,1) and ϕ include the entire set of variables of the system under consideration.
The Schr\"odinger equation for the two-dimensional simple harmonic oscillator is solved using elliptic coordinates where it is separable. The separability of the problem in such coordinates is independent of the selection of the focal distance. The solutions are labeled by the total number of quanta N and by a set of characteristic values b corresponding to the eigenvalues of an observable B^, which does not commute with L^, the total angular momentum or H${^}_{x}$, the energy associated with the x degree of freedom. The well-known quantum energies as well as the characteristic values are obtained by imposing physical polynomial solutions.