Finite-element lattice Hamiltonian matrix elements: Anharmonic oscillators
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The finite-element approach to lattice field theory is both highly accurate (relative errors ∼1/N 2, 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=p 2/2+λq 2k /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.
- BenderC. M., MeadL. R. and MiltonK. A.,Comput. Math. Appl. 28, 279 (1994).
- MiltonK. A. and GroseT.,Phys. Rev. D 41, 1261 (1990).
- 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.
- MillerD., MiltonK. A. and Siegemund-BrokaS.,Phys. Rev. D 46, 806 (1993).
- 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.
- MiltonK. A., Absence of species doubling in finite-element quantum electrodynamics,Lett. Math. Phys. 34, 285–295 (1995).
- BenderC. M., MiltonK. A., SharpD. H., SimmonsL. M.Jr., and StongR.,Phys. Rev. D 32, 1476 (1985).
- BenderC. M., SimmonsL. M.Jr. and StongR.,Phys. Rev. D 33, 2362 (1986).
- 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.
- BenderC. M. and GreenM. L.,Phys. Rev. D. 34, 3255 (1986).
- StevensonP. M.,Phys. Rev. D 23, 2916 (1981).
- 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.
- BenderC. M. and OrzagS. A.,Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978, p. 523.
- BenderC. M. and MiltonK. A.,Phys. Rev. D. 34, 3149 (1986).
- Finite-element lattice Hamiltonian matrix elements: Anharmonic oscillators
Letters in Mathematical Physics
Volume 36, Issue 2 , pp 177-187
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Primary: 81Q99
- Secondary: 39A10
- lattice field theory
- finite-element method
- Gaussian knots
- quantum systems
- matrix elements
- Author Affiliations
- 1. Department of Physics and Astronomy, The University of Oklahoma, 73019, Norman, OK, USA
- 2. Theoretical Physics Group, Blackett Laboratory, Imperial College, Prince Consort Road, SW7 2BZ, London, UK
- 3. Oklahoma School of Science and Mathematics, 1141 Lincoln Blvd., 73104, Oklahoma City, OK, USA