Semiclassical behavior at a quantum avoided crossing

Joyeux, Marc
February 1995
Journal of Chemical Physics;2/15/1995, Vol. 102 Issue 7, p2816
Academic Journal
For a polynomial potential with resonant fundamental frequencies (1:2 and 1:3 resonances), quantum avoided crossings can occur when quantum eigenvalues are plotted versus a parameter in the Hamiltonian. In the present paper, primitive (EBK) semiclassical behavior at the quantum avoided crossing is reinvestigated, using the exact analytical calculation of the action integrals, which was devised recently [Chem. Phys. 185, 263 (1994)] for an approximate resonance Hamiltonian that can be deduced from the exact polynomial Hamiltonian by low order perturbation theory. The previously reported behavior, that is semiclassical levels passing through the intersection instead of avoiding each other, is shown to happen if there exist two superimposed branches in the plot of the second action integral I2 as a function of the energy. These results are interpreted in terms of semiclassical diabatic basis and of quantum dynamical tunneling. In contrast, if the semiclassical system enters the (anti)crossing region with semiclassical quantum numbers I2 which do not lie on superimposed branches of the plot, it is shown that at least one, and possibly two, level(s) must cross the separatrix, that is pass from the inside to the outside of the resonance region (or conversely) in order to adapt to the quantum avoided crossing. This causes (i) corresponding semiclassical quantum number I2 to change (ii) the close correspondence between quantum and semiclassical mechanics to break down. © 1995 American Institute of Physics.


Related Articles

  • Adiabatic evolution for systems with infinitely many eigenvalue crossings. Joye, A.; Monti, F.; Guerin, S.; Jauslin, H.R. // Journal of Mathematical Physics;Nov99, Vol. 40 Issue 11, p5456 

    Focuses on the formulation of an adiabatic theorem for Hamiltonian systems with infinitely many eigenvalue crossings. Availability of intense pulsed laser sources; Description of the interaction of the Hamiltonian molecule; Review of the adiabatic approximation in quantum mechanics.

  • The doublet representation of non-Hilbert eigenstates of the Hamiltonian. Castagnino, M.; Domenech, G.; Levinas, M.; Umerez, N. // Journal of Mathematical Physics;May96, Vol. 37 Issue 5, p2107 

    Studies the minimal mathematical structure used to represent quantum eigenstates with complex eigenvalues without the need of analytic continuation. Use of the eigenvectors to build doublets in non-Hilbert spaces; Solutions for the Friedrich model that continuously join the ones of free...

  • An inversion inequality for potentials in quantum mechanics. Hall, Richard L. // Journal of Mathematical Physics;May99, Vol. 40 Issue 5, p2254 

    Focuses on the inversion inequality for potentials in quantum mechanics. Supposition that the ground-state eigenvalue E=F(v) of the Schrodinger Hamiltonian in one dimension is known for all values of the coupling v>0; Expression of the potential shape in the form f(x)=G(x[sup 2]); Establishment...

  • The 4-particle hydrogen-antihydrogen system revisited. Van Hooydonk, G. // European Physical Journal D -- Atoms, Molecules, Clusters & Opti;Mar2005, Vol. 32 Issue 3, p299 

    The historical importance of the original quantum mechanical bond theory proposed by Heitler and London in 1927 as well as its pitfalls are reviewed. Modern ab initio treatments of H-systems are inconsistent with the logic behind algebraic Hamiltonians H± =H 0 ± ?H...

  • Quaternionic Formulation of Supersymmetric Quantum Mechanics. Rawat, Seema; Negi, O. P. S. // International Journal of Theoretical Physics;Feb2009, Vol. 48 Issue 2, p305 

    Quaternionic formulation of supersymmetric quantum mechanics has been developed consistently in terms of Hamiltonians, super partner Hamiltonians, and supercharges for free particle and interacting field in one and three dimensions. Supercharges, super partner Hamiltonians and energy eigenvalues...

  • Bosonic quasideterminants and eigenvalue problems of generalized spin-orbit operators. Ben Geloun, Joseph; Hounkonnou, M. Norbert // Journal of Mathematical Physics;Feb2008, Vol. 49 Issue 2, p023509 

    This paper deals with an extension of the applications of the paper by Gelfand and Retakh [Funct. Anal. Appl. 25, 91 (1991)] on quasideterminant (QsD) algebraic method to eigenvalue problems in quantum mechanics. Using relevant identities on the free 1-mode bosonic algebra, we build...

  • Lower spectral branches of a spin-boson model. Angelescu, Nicolae; Minlos, Robert A.; Ruiz, Jean; Zagrebnov, Valentin A. // Journal of Mathematical Physics;Oct2008, Vol. 49 Issue 10, p102105 

    We study the structure of the spectrum of a two-level quantum system weakly coupled to a boson field (spin-boson model). Our analysis allows to avoid the cutoff in the number of bosons, if their spectrum is bounded below by a positive constant. We show that, for small coupling constant, the...

  • Real Eigenvalue of a Non-Hermitian Hamiltonian System. Singh, Ram Mehar // Applied Mathematics;Oct2012, Vol. 3 Issue 10, p1117 

    With a view to getting further insight into the solutions of one-dimensional analogous Schrödinger equation for a non-hermitian (complex) Hamiltonian system, we investigate the quasi-exact PT- symmetric solutions for an octic potential and its variant using extended complex phase space...

  • Spectra of Photon Down Conversion. Smotlacha, J.; Chadzitaskos, G.; Daskaloyannis, C. // AIP Conference Proceedings;11/30/2009, Vol. 1191 Issue 1, p166 

    We demonstrate that quasi-exactly solvable models of quantum mechanics can be used in nonlinear optical processes for a down conversion or second-harmonic generation processes. After the reduction we use the Turbiner and Bender -Dunne polynomial approach. The eigenvalues of Hamiltonian for low...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics