Letters in Mathematical Physics

, Volume 36, Issue 2, pp 177–187

Finite-element lattice Hamiltonian matrix elements: Anharmonic oscillators

  • Kimball A. Milton
  • Rhiju Das
Article

Abstract

The finite-element approach to lattice field theory is both highly accurate (relative errors ∼1/N2, whereN is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this Letter, we construct matrix elements for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian isH=p2/2+λq2k/2k. Construction of such matrix elements does not require solving the implicit equations of motion. Low-order approximations turn out to be quite accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator groundstate gives a result for thek=2 anharmonic oscillator groundstate energy accurate to better than 1% while a two-state approximation reduces the error to less than 0.1%. Accurate wavefunctions are also extracted. Analogous results may be obtained in the continuum, but there the computation is more difficult, and not generalizable to field theories in more dimensions.

Mathematics Subject Classifications (1991)

Primary: 81Q99 Secondary: 39A10 

Key words

lattice field theory Hamiltonians finite-element method Gaussian knots quantum systems matrix elements 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    BenderC. M., MeadL. R. and MiltonK. A.,Comput. Math. Appl. 28, 279 (1994).Google Scholar
  2. 2.
    MiltonK. A. and GroseT.,Phys. Rev. D 41, 1261 (1990).Google Scholar
  3. 3.
    MiltonK. A., in K. K.Phua and Y.Yamaguchi (eds),Proc. XXVth Internat. Conf. on High-Energy Physics, Singapore, 1990, World Scientific, Singapore, 1991, p. 432.Google Scholar
  4. 4.
    MillerD., MiltonK. A. and Siegemund-BrokaS.,Phys. Rev. D 46, 806 (1993).Google Scholar
  5. 5.
    Miller, D., Milton, K. A. and Siegemund-Broka, S., Finite-element quantum electrodynamics. II. Lattice propagators, current commutators, and axial-vector anomalies, Preprint OKHEP-93-11, hep-ph/9401205, submitted toPhys. Rev. D. Google Scholar
  6. 6.
    MiltonK. A., Absence of species doubling in finite-element quantum electrodynamics,Lett. Math. Phys. 34, 285–295 (1995).Google Scholar
  7. 7.
    BenderC. M., MiltonK. A., SharpD. H., SimmonsL. M.Jr., and StongR.,Phys. Rev. D 32, 1476 (1985).Google Scholar
  8. 8.
    BenderC. M., SimmonsL. M.Jr. and StongR.,Phys. Rev. D 33, 2362 (1986).Google Scholar
  9. 9.
    Milton, K. A., Finite-element time evolution operator for the anharmonic oscillator, Preprint OKHEP-94-01, hep-ph/9404286, to appear inProc. Harmonic Oscillators II, Cocoyoc, Mexico, 23–25 March, 1994.Google Scholar
  10. 10.
    BenderC. M. and GreenM. L.,Phys. Rev. D. 34, 3255 (1986).Google Scholar
  11. 11.
    StevensonP. M.,Phys. Rev. D 23, 2916 (1981).Google Scholar
  12. 12.
    BarnesJ. F., BrascampH. J. and LiebE. H., in E. H.Lieb, B.Simon, and A. S.Wightman (eds),Studies in Mathematical Physics, Princeton University Press, Princeton, NJ, 1976.Google Scholar
  13. 13.
    BenderC. M. and OrzagS. A.,Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978, p. 523.Google Scholar
  14. 14.
    BenderC. M. and MiltonK. A.,Phys. Rev. D. 34, 3149 (1986).Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Kimball A. Milton
    • 1
    • 2
  • Rhiju Das
    • 3
  1. 1.Department of Physics and AstronomyThe University of OklahomaNormanUSA
  2. 2.Theoretical Physics Group, Blackett LaboratoryImperial CollegeLondonUK
  3. 3.Oklahoma School of Science and MathematicsOklahoma CityUSA

Personalised recommendations