My Research Interests

My research interests lie in the area of semiclassical mechanics, as applied to physical systems. To implement these methods, it is necessary to calculate classical trajectories (solutions to Hamilton's equations), and perform integrations over multidimensional integrands. One invariably encounters chaotic regions of phase space in these calculations, which introduce considerable numerical difficulties. Therefore, my research involves a multitude of fields, including:

  • the classical limit of quantum mechanics
    • Wigner transform
    • Moyal quantum mechanics
  • dynamical systems
    • Initial value problems (Hamilton's equations)
    • chaos
    • fixed points and periodic trajectories
  • numerical integration
    • multidimensional integrals
    • rapidly oscillating integrands
    • Monte Carlo integration
  • atomic and molecular scattering
    • collisional interference HD-rare gas systems
  • molecular photodissociation
    • collinear CO2 cross sections (total)
    • direct part of photodissociation cross sections (total and partial)

I am also interested in solving the Schrödinger equation using quantum perturbation theory. This leads to investigations of divergent series, and attempts to assign a sum to the divergent series using Pade approximants, Borel resummation, continued fractions, and other techniques.

If any portion of this interests you, or you think it might be interesting but you really aren't sure, stop by for a chat. More detailed descriptions of my research projects follow.


Semiclassical Time-Dependent Propagators

If an initial wave function for a system is known, the evolution of the wave function in time for the system can be predicted if the quantum mechanical time-dependent propagator is known. In many cases, the quantum propagator may be approximated by a semiclassical propagator. These propagators are relatively simple to compute as they are expressed as phase space integrals. However, for chaotic systems the rapid divergence of classical trajectories results in integrals which can not be determined accurately without calculating an excessive number of trajectories.

I have completed a survey of a number of techniques which have been introduced to allow traditional semiclassical propagators to treat chaotic systems; specifically, to calculate the Franck-Condon spectrum of model two dimensional systems. Currently, I am engaged in extending the analysis from the relatively simple (with its reliance on overlap with the initial wavefunction) Franck-Condon spectrum to the determination of photodissociation matrix elements. If it proves feasible to calculate these quantities semiclassically, this could open the door to a semiclassical based theory of coherent control.

The technique I have settled on as most appropriate for numerical computation involves the semiclassical Herman-Kluk propagator. The derivation of this propagator is heuristic in nature, and if it proves useful to theoretical chemists and physicists I would attempt a rigorous derivation of this propagator from the Feynman propagator.

  • B. R. McQuarrie, Dmitri G. Abrashkevich, Paul Brumer (2003) Classical-Wigner Phase Space Approximation to Cumulative Matrix Elements in Coherent Control, J. Chem. Phys. 119, 3606--3618. link to article (arXiv.org)
  • McQuarrie, B.R., and Brumer, Paul (2000) Semiclassical initial value representation techniques for chaotic systems, Chemical Physics Letters 319: 27-44. link to article (science direct)

Moyal Quantum Mechanics

Moyal quantum mechanics is a semiclassical representation which gives a complete description of quantum mechanics based on classical phase space quantities. Quantities are expressed in terms of expansions in Planck's constant, and to infinite order the description is fully quantum in its realization. To zeroth order in Planck's constant the description is essentially classical, although quantum ordering effects still remain. Expanding quantities to consistent order in Planck's constant allows one to measure quantum effects in a system.

Historically, Moyal quantum mechanics has had little interaction with real physical systems. I have worked closely with Prof. T. Osborn and Prof. G. Tabisz to develop the theory of Moyal quantum mechanics to a level which can be used to treat physically meaningful systems. I was solely responsible for the conversion of this complex theory into a workable fortran code to compute expectation values to second order in Planck's constant. The system studied was a three dimensional Gaussian elastically scattered by a Lennard-Jones potential.

Many spectral line shape problems rely on what is known as the classical path approximation to determine interaction dynamics. The Moyal formalism has recently been extended by Prof. Tabisz, Prof. Osborn, Dr. Kondrat'eva and myself to treat composite systems--systems which split into a number of different subsystems with independent degrees of freedom. This Moyal composite formalism can be used to treat spectral line shape problems. In this context, a mathematically rigorous semiclassical spectral line shape theory has been developed which contains the classical path approximation as a first order result.

  • Osborn, T.A., Kondrat'eva, M.F., Tabisz G.C., and McQuarrie, B.R. (1999) Mixed Weyl symbol calculus and spectral line shape theory, Journal of Physics A: Mathematical and General 32: 4149-4169. link to article (arXiv.org)
  • McQuarrie, B.R., Osborn, T.A., and Tabisz, G.C. (1998) Semiclassical Moyal Quantum Mechanics for Atomic Systems, Physical Review A. 58: 2944-2961. link to article (pra)

Spectral Line Shape Calculations

I have investigated the role that induced dipole components play in collisional interference in the rotational spectrum of HD-rare gas systems. The theory was also extended to treat vibrational transitions for the spectrum of HD-He. The classical path approximation is used in this theory, and was the motivating factor in proceeding to the Moyal quantum mechanics analysis.

  • McQuarrie, B.R., and Tabisz, G.C. (1996) Collisional Interference in the Infrared Spectrum of HD: Calculation of the Line Shape of Vibrational Transitions for HD-He. Journal of Molecular Liquids. 70: 159-168. link to article (science direct)
  • McQuarrie, B.R., Tabisz, G.C., Gao, B., and Cooper J. (1995) Role of the induced dipole moment in the collisional interference in the pure rotational spectrum of HD-He and HD-Ar. Physical Review A. 52: 1976-1981. link to article (pra)

The Schrödinger Equation

I have investigated the application of the hypervirial and Hellmann-Feynman theorems to the perturbed radial Klein-Gordon equation with Coulomb interaction. This generates a divergent series for the energy eigenvalue, which is summed using continued fractions. The computer code to implement this procedure was written in the symbolic language maple.

  • McQuarrie, B.R., and Vrscay, E.R. (1993) Rayleigh-Schrödinger Perturbation Theory at Large Order for Radial Klein-Gordon Equations. Physical Review A. 47: 868-875. link to article (pra)

Office Hours Sci 1380:
Drop-in Office Hours (no appointment needed) are listed on google calendar.

Appointment:
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Email:
mcquarrb@morris.umn.edu

Phone:
589-6302
(I do not use voicemail)