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International Journal of Theoretical Physics

The Born–Oppenheimer Approximation From the $$\textsf{QU}$$ Decomposition of Bilinearly Coupled Harmonic Oscillators

Carlos A. Arango

Published Sep 8, 2026• 0 citations
Publisher

Abstract

Abstract For two bilinearly coupled harmonic oscillators, whose exact solution is known analytically, the linear transformation from physical to normal-mode coordinates is obtained explicitly. This transformation belongs to $$\textbf{GL}(\textbf{2},\mathbb {R})$$ GL ( 2 , R ) and admits a unique $$\textsf{QU}$$ QU (Iwasawa) factorization into an orthogonal component describing symmetric mixing and an upper-triangular component encoding a hierarchical dependence of fast on slow coordinates. The Born–Oppenheimer approximation is identified with this triangular factor. In the adiabatic limit of small frequency ratio, the orthogonal contribution becomes negligible, and the exact transformation reduces to the Born–Oppenheimer map. This identification is supported by numerical comparisons of energies and wavefunctions across different coupling regimes.

DOI
10.1007/s10773-026-06476-1

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The Born–Oppenheimer Approximation From the QU\textsf{QU}QU Decomposition of Bilinearly Coupled Harmonic Oscillators
Abstract For two bilinearly coupled harmonic oscillators, whose exact solution is known analytically, the linear transformation from physical to normal-mode coordinates is obtained explicitly. This transformation belongs to GL(2,R)\textbf{GL}(\textbf{2},\mathbb {R})GL(2,R) GL ( 2 , R ) and admits a unique QU\textsf{QU}QU QU (Iwasawa) factorization into an orthogonal component describing symmetric mixing and an upper-triangular component encoding a hierarchical dependence of fast on slow coordinates. The Born–Oppenheimer approximation is identified with this triangular factor. In the adiabatic limit of small frequency ratio, the orthogonal contribution becomes negligible, and the exact transformation reduces to the Born–Oppenheimer map. This identification is supported by numerical comparisons of energies and wavefunctions across different coupling regimes.
The nature of the adiabatic approximation is investigated. This approximation is at the center of the Born-Oppenheimer approximation, commonly employed in the construction of molecular potentials. First, general behaviors of the approximation and some implications of these results are discussed. We show that the ground-state energy, calculated using the adiabatic approximation, will be lower than the calculated energy when no approximations are made. Next, a numerical and analytical application of the adiabatic approximation to systems comprising two coupled harmonic oscillators is described. These systems are used to demonstrate the results derived in the first part of the paper and to investigate how the accuracy of the approximation depends on the frequencies of the oscillators and the coupling strength.
Considering classical environment to be a classical oscillator, we describe the dynamics of a system consisting of a quantum oscillator coupled to classical oscillator. Using the Born-Oppenheimer approximation and the method of adiabatic invariants, we derive the classically effective Hamiltonian describing the complete dynamics of the system and the wavefunction of quantum part of the system. We also present a new method for investigating the dynamics of the quantum system in classical environment.
In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.
Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.
An exact invariant operator of time-dependent coupled oscillators is derived using the Liouville-von Neumann equation. The unitary relation between this invariant and the invariant of two uncoupled simple harmonic oscillators is represented. If we consider the fact that quantum solutions of the simple harmonic oscillator is well-known, this unitary relation is very useful in clarifying quantum characteristics of the original systems, such as entanglement, probability densities, fluctuations of the canonical variables, and decoherence. We can identify such quantum characteristics by inversely transforming the mathematical representations of quantum quantities belonging to the simple harmonic oscillators. As a case in point, the eigenfunctions of the invariant operator in the original systems are found through inverse transformation of the well-known eigenfunctions associated with the simple harmonic oscillators.
The Born-Oppenheimer (BO) and single-surface Born-Huang (BH) approximations replace the full molecular problem by nuclear motion associated with one clamped electronic state, whereas the full BH expansion and exact factorization (EF) are exact representations. We use two analytically transparent models to identify what controls one-surface error. For Fernández's bilinearly coupled oscillators, the Brattsev ground-state bracket was already known; we derive closed-form differences proving strictness for every positive nuclear mass and nonzero stable coupling. Using Hunter's conditional-amplitude construction, we place the closed-form ground-state EF potential alongside BO and single-surface BH. Exact excited-state calculations show that the bracket does not extend through the spectrum. The diagonal BH correction is the quantum-metric coefficient of the retained electronic state divided by twice the nuclear mass. Fernández's sixth-order expansion shows that the next omitted correction contains the same derivative couplings, an additional inverse electronic-energy separation, and a factor depending on the nuclear state. We therefore add a two-state model with a position-dependent gap. Decreasing its minimum separation makes the metric higher and narrower while leaving the total Fubini-Study length equal to pi/2. The exact EF, BO, and BH potentials can differ visibly even when their nuclear ground-state amplitudes are nearly identical. The two models show that one-surface accuracy depends on nuclear mass, electronic-state variation, energy separation from omitted states, and the nuclear wavefunction.