The use of the BornOppenheimer factorization in the phasespace representation of the timeindependent Schrödinger equation for bilinearly coupled harmonic oscillators
Abstract
A system of two bilinearly coupled harmonic oscillators has been solved analytically by using the BornOppenheimer (BO) product wavefunction ansatz and the phasespace bound trajectory approach [J. S. Molano et al., Chem. Phys. Lett. \textbf{76}(12), 138171 (2021)]. The bilinearly coupled oscillator system allows to obtain the analytical expression of the quantum system, facilitating comparison with the results of using the BO ansatz product. The analytical and BO wavefunctions are obtained as a product of parabolic cylinder functions. The arguments of the parabolic cylinder functions of the exact and the BO wavefunctions are related to the physical coordinates by linear transformations, represented by matrices $\mathsf{G}$ and $\mathsf{\tilde{G}}$ respectively. A $QU$ decomposition of the matrix $\mathsf{G}$ outputs an upper triangular matrix $\mathsf{U}$ that is closely related to the BO $\mathsf{\tilde{G}}$ matrix. The effect of the BO nonadiabatic coupling is analyzed on the stability of the phasespace equations of motion. The eigenvalues and eigenfunctions obtained by the BornOppenheimer phasespace trajectory approach show excellent agreement with the analytical solutions.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.11500
 Bibcode:
 2021arXiv210811500A
 Keywords:

 Quantum Physics;
 Physics  Chemical Physics
 EPrint:
 23 pages, 5 figures