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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Discrete-time quantum mechanics

Carl M. Bender, Kimball A. Milton, David H. Sharp, L. M. Simmons, Richard Stong

Published Sep 15, 1985• 55 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.32.1476

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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1986•8 citations
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.
In a previous paper we used the method of finite elements to formulate consistent, unitary, discrete-time quantum-mechanical systems having one degree of freedom. In this paper we extend the treatment to systems having two degrees of freedom. The proof of consistency is more delicate: It is nontrivial to show that there are no operator-ordering problems and that independent degrees of freedom remain independent at subsequent lattice sites. The construction of purely bosonic lattice systems is a straightforward application of the finite-element prescription. However, it is surprising that quantum-mechanical systems having interacting fermions and bosons are only unitary if the interaction is modified on the lattice by a term which vanishes as the lattice spacing approaches zero. The modified interaction is determined by an interesting nonlinear condition. In all cases we give an explicit formula for the lattice transfer operator.
The finite-element approach to lattice field theory is both highly accurate (relative errors ∼1/N2\sim1/N^2∼1/N2, where NNN 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=p2/2+λq4/4.H=p^2/2+λq^4/4.H=p2/2+λq4/4. 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\%.
Inspired by the discrete evolution implied by the recent work on loop quantum cosmology, we obtain a discrete time description of usual quantum mechanics viewing it as a constrained system. This description, obtained without any approximation or explicit discretization, mimics features of the discrete time evolution of loop quantum cosmology. We discuss the continuum limit, physical inner product and matrix elements of physical observables to bring out various issues regarding the viability of a discrete evolution. We also point out how a continuous time could emerge without appealing to any continuum limit.
Inspired by the discrete evolution implied by the recent work on loop quantum cosmology, we obtain a discrete time description of usual quantum mechanics viewing it as a constrained system. This description, obtained without any approximation or explicit discretization, mimics features of the discrete time evolution of loop quantum cosmology. We discuss the continuum limit, physical inner product and matrix elements of physical observables to bring out various issues regarding viability of a discrete evolution. We also point out how a continuous time could emerge without appealing to any continuum limit.