Quantum roll: A study of the long-time behavior of the finite-element method

Journal Article · Tue Oct 15 00:00:00 EDT 1985 · Phys. Rev. D; (United States)

DOI:https://doi.org/10.1103/PhysRevD.32.2056· OSTI ID:5201144

Cooper, F; Milton, K A; Simmons, Jr, L M

Using the method of finite elements we investigate the quantum behavior of a particle starting in an unstable equilibrium at the top of a potential hill and rolling down. In order to study the numerical accuracy of the method for large times we consider the exactly solvable model with V(q) = -1/2q/sup 2/ and an initial Gaussian wave function at t = 0, psi(q)proportionalexp(-1/2q/sup 2/) so that the initial state Vertical Bar0> has <0Vertical Barq/sup 2/Vertical Bar0> = <0Vertical Barp/sup 2/Vertical Bar0> = 1/2. We study the accuracy of the large-time approximations to <0Vertical Barq/sup 2/(t)Vertical Bar0> based upon single finite elements of degree n. The Taylor series is exact up to t/sup 2n/ and even the coefficient of t/sup 2n/+2 is a very accurate representation of the exact coefficient. We then consider the convergence properties of the (N,n) approximations consisting of N iterations of the finite element of degree n. For these approximations the corrections to the Taylor-series coefficients in higher orders vanish as N/sup -2n/.

• Research Organization: Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
• OSTI ID: 5201144
• Journal Information: Phys. Rev. D; (United States), Vol. 32:8
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
HARMONIC OSCILLATORS
FINITE ELEMENT METHOD
FUNCTIONALS
HEISENBERG PICTURE
MONTE CARLO METHOD
QUANTUM MECHANICS
SPACE-TIME
WAVE EQUATIONS
DIFFERENTIAL EQUATIONS
ELECTRONIC EQUIPMENT
EQUATIONS
EQUIPMENT
FUNCTIONS
MECHANICS
NUMERICAL SOLUTION
OSCILLATORS
PARTIAL DIFFERENTIAL EQUATIONS
657002* - Theoretical &amp; Mathematical Physics- Classical &amp; Quantum Mechanics