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arXiv:hep-ph/9404286

Finite-Element Time Evolution Operator for the Anharmonic Oscillator

Kimball A. Milton

Published Apr 18, 1994• 1 citation

Abstract

The finite-element approach to lattice field theory is both highly accurate (relative errors $\sim1/N^2$, where $N$ 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 talk I construct matrix elements for dynamical variables and for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian is $H=p^2/2+λq^4/4.$ Construction of such matrix elements does not require solving the implicit equations of motion. Low order approximations turn out to be extremely accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator ground state gives a result for the anharmonic oscillator ground state energy accurate to better than 1\%, while a two-state approximation reduces the error to less than 0.1\%.

DOI: 10.48550/arxiv.hep-ph/9404286

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The finite-element approach to lattice field theory is both highly accurate (relative errors

\sim1/N^2

, where

N

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 talk I construct matrix elements for dynamical variables and for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian is

H=p^2/2+λq^4/4.

Construction of such matrix elements does not require solving the implicit equations of motion. Low order approximations turn out to be extremely accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator ground state gives a result for the anharmonic oscillator ground state energy accurate to better than 1%, while a two-state approximation reduces the error to less than 0.1%.

For an anharmonic oscillator, it follows from the number of zeros of the exact wave functions as functions of the quantum number

n

that the matrix elements $〈n|x|{n}^{\ensuremath{'}}〉$ and $〈n|p|{n}^{\ensuremath{'}}〉$ should be rapidly decreasing functions of $|n\ensuremath{-}{n}^{\ensuremath{'}}|$. Matrix elements of polynomials in

x

and

p

should therefore be well approximated by a finite number of terms in their sum-rule decomposition. From the matrix elements of the equations of motion for

x

and for

p

and of the commutator

|x,p|

, one thereby obtains closed sets of nonlinear algebraic equations to characterize subspaces of the Hilbert space of exact eigenfunctions. The approximations are also derived from a novel variational principle, and numerous variant approximation schemes are suggested. Essentially exact numerical results are obtained and compared with previous work. The broad applicability of the techniques is emphasized.

This is an investigation of an anharmonic oscillator characterized by the potential (ω02/ 2) q2 + λ′ q4. By using the exact relations obtained by evaluating ≪i|[H, O]∓|j > where O is an arbitrary operator, H is our total Hamiltonian and |i > and |j > are exact eigenstates of H, we derive an exact recurrence formula. This formula allows us to express τ-functions with a higher power of the variables through τ-functions with a lower power of the variables and energy eigenvalues. In this way we derive several exact relations, which are, in a sense, generalizations of the virial theorem and sum rules (Eqs. (4·5), (4·6) and (5·4)). These exact relations are the central equaions of this paper. On the basis of these exact relations we propose our “nearest neighbour level” (N. N. L) approximation, which seems to provide a good approximation scheme. We can also use our exact relations to test the validity of various approximation methods, and as an example, we discuss the “New-Tamm-Dancoff” (N. T. D)-type of approximation in detail.

The method of finite elements converts the operator Heisenberg equations that arise from a Hamiltonian of the form

H=\frac{{p}^{2}}{2}+V(q)

into a set of operator difference equations on a lattice. The equal-time commutation relations are exactly preserved and thus are consistent with the requirements of unitarity. We consider general Hamiltonians of the form

H(p,q)

and show that the requirement of unitarity uniquely determines the operator ordering in such Hamiltonians. (The ordering procedure involves a set of orthogonal polynomials which are not widely known.) Our result shows that it is possible to treat quantum spin systems by the method of finite elements.

The finite-element (collocation) method enables us to construct discrete time-lattice quantum systems that accurately approximate continuum quantum systems. The discrete quantum systems so generated are fully consistent quantum-mechanical systems in their own right. This paper gives a comprehensive treatment of such quantum systems. We examine various finite-element schemes, construct the effective lattice Hamiltonian, and calculate eigenvalues. Numerical results are extremely easy to obtain and are very accurate.