We study the application of the "leapfrog" method of finite differencing to the approximate solution of operator equations of motion in quantum theory. We show that, for a wide class of linear and nonlinear systems, the leapfrog differencing scheme is exactly unitary. The method is sufficiently general to apply to many-particle systems with arbitrary potential forces and to lattice-regulated nonlinear $\ensuremath{\sigma}$ models and non-Abelian gauge theories. In contrast to the recent proposal of Bender and Sharp (which is based on the finite-elements method) our approach is explicit rather than implicit and, in the case of lattice-regulated field theories, has a lattice analog of microcausality. For systems with finitely many degrees of freedom and self-adjoint Hamiltonians, we show that our approximate solutions converge to the exact solutions in the limit in which the time step tends to zero.We study the application of the "leapfrog" method of finite differencing to the approximate solution of operator equations of motion in quantum theory. We show that, for a wide class of linear and nonlinear systems, the leapfrog differencing scheme is exactly unitary. The method is sufficiently general to apply to many-particle systems with arbitrary potential forces and to lattice-regulated nonlinear $\ensuremath{\sigma}$ models and non-Abelian gauge theories. In contrast to the recent proposal of Bender and Sharp (which is based on the finite-elements method) our approach is explicit rather than implicit and, in the case of lattice-regulated field theories, has a lattice analog of microcausality. For systems with finitely many degrees of freedom and self-adjoint Hamiltonians, we show that our approximate solutions converge to the exact solutions in the limit in which the time step tends to zero.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsNov 15, 198346 citations

Finite element descriptions of quantum mechanical systems

Holly S. Randa

Jan 1, 19960 citations

Matrix mechanics as a practical tool in quantum theory: The anharmonic oscillator

Ching‐Teh Li, Abraham Klein, Franz R. Krejs

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.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.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsOct 15, 197513 citations

Quantum-mechanical tunneling and finite elements

R. Goloskie, J. W. Kramer, L. R. Ram‐Mohan

Computers in PhysicsNov 1, 19944 citations

Pairing in Finite Systems: Beyond the HFB Theory

L. M. Robledo, G. F. Bertsch

The Hartree-Fock-Bogoliubov approximation is very useful for treating both long- and short-range correlations in finite quantum fermion systems, but it must be extended in order to describe detailed spectroscopic properties. One problem is the symmetry-breaking character of the HFB approximation. We present a general and systematic way to restore symmetries and to extend the configuration space using pfaffian formulas for the many-body matrix elements. The advantage of those formulas is that the sign of the matrix elements is unambiguously determined. It is also helpful to extend the space of configurations by constraining the HFB solutions in some way. A powerful method for finding these constrained solutions is the gradient method, based on the generalized Thouless transformation. The gradient method also preserves the number parity of the Bogoliubov transformation, which facilitates the application of the theory to systems with odd particle number.The Hartree-Fock-Bogoliubov approximation is very useful for treating both long- and short-range correlations in finite quantum fermion systems, but it must be extended in order to describe detailed spectroscopic properties. One problem is the symmetry-breaking character of the HFB approximation. We present a general and systematic way to restore symmetries and to extend the configuration space using pfaffian formulas for the many-body matrix elements. The advantage of those formulas is that the sign of the matrix elements is unambiguously determined. It is also helpful to extend the space of configurations by constraining the HFB solutions in some way. A powerful method for finding these constrained solutions is the gradient method, based on the generalized Thouless transformation. The gradient method also preserves the number parity of the Bogoliubov transformation, which facilitates the application of the theory to systems with odd particle number.

arXiv:1205.4443WORLD SCIENTIFIC eBooksMar 1, 20133 citations

Finite Element Quantum Mechanical Analysis

Nobuhiro YOSHIKAWA, Yasuyuki Mitsui, Osamu Kuwazuru +1 more

Finite Element Method， O ( N ) Methods

Proceedings of the 1992 Annual Meeting of JSME/MMDJan 1, 20030 citations

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.