Physical Review Letters

Resolution of the Operator-Ordering Problem by the Method of Finite Elements

Carl M. Bender, Lawrence R. Mead, Stephen S. Pinsky

Published Jun 9, 1986• 42 citations

Abstract

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.

DOI: 10.1103/physrevlett.56.2445

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Physical Review Letters

Resolution of the Operator-Ordering Problem by the Method of Finite Elements

Carl M. Bender, Lawrence R. Mead, Stephen S. Pinsky

Published Jun 9, 1986• 42 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevlett.56.2445

Related papers

Finite-Element Time Evolution Operator for the Anharmonic Oscillator

Kimball A. Milton

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\%.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\%.

arXiv:hep-ph/9404286Apr 18, 19941 citation

The Quantum-Mechanical Equations of Motion and Commutation Relations

Calvin R. Putnam

It has been shown by Wigner that the equations of motion of a quantum-mechanical system with a hamiltonian $H=\frac{1}{2}{p}^{2}+V(q)$ need not imply the commutation relation $i[p,q]=1$ . Particular examples are furnished by the free particle and the harmonic oscillator. On the other hand, Wigner points out that the above commutation rule does follow from the equations of motion if $V(q)$ is of the form $V=aq+b$ or $V=a{q}^{3}$ . In the present note, this latter result is generalized to the case of an arbitrary potential $V=a{q}^{n}+b$ , where $n$ is an odd integer; the last-named potentials are, in fact, particular cases of those satisfying a general criterion contained in the article.It has been shown by Wigner that the equations of motion of a quantum-mechanical system with a hamiltonian $H=\frac{1}{2}{p}^{2}+V(q)$ need not imply the commutation relation $i[p,q]=1$. Particular examples are furnished by the free particle and the harmonic oscillator. On the other hand, Wigner points out that the above commutation rule does follow from the equations of motion if $V(q)$ is of the form $V=aq+b$ or $V=a{q}^{3}$. In the present note, this latter result is generalized to the case of an arbitrary potential $V=a{q}^{n}+b$, where $n$ is an odd integer; the last-named potentials are, in fact, particular cases of those satisfying a general criterion contained in the article.

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FINITE-DIFFERENCE EQUATIONS IN RELATIVISTIC QUANTUM MECHANICS 1

V. Aldaya, Julio Guerrero

Relativistic Quantum Mechanics suffers from structural problems which are traced back to the lack of a position operator ˆx, satisfying [ˆx, ˆp] = i¯hˆ1 with the ordinary momentum operator ˆp, in the basic symmetry group – the Poincaré group. In this paper we provide a finite-dimensional extension of the Poincaré group containing only one more (in 1+1D) generator ˆπ, satisfying the commutation relation [ ˆ k, ˆπ] = i¯hˆ1 with the ordinary boost generator ˆ k. The unitary irreducible representations are calculated and the carrier space proves to be the set of Shapiro’s wave functions. The generalized equations of motion constitute a simple example of exactly solvable finite-difference set of equations associated with infinite-order polarization equations. Introduction: Higher-order Polarizations One essential difference between Geometric Quantization [1, 2, 3, 4] and a Group Approach to Quantization (GAQ) [5, 6] is the possibility of introducing in the latter higher-order polarizations made out of elements of the (left) enveloping algebra. These higher-order polarizations are especially suitable for those (anomalous) cases in which theRelativistic Quantum Mechanics suffers from structural problems which are traced back to the lack of a position operator ˆx, satisfying [ˆx, ˆp] = i¯hˆ1 with the ordinary momentum operator ˆp, in the basic symmetry group – the Poincaré group. In this paper we provide a finite-dimensional extension of the Poincaré group containing only one more (in 1+1D) generator ˆπ, satisfying the commutation relation [ ˆ k, ˆπ] = i¯hˆ1 with the ordinary boost generator ˆ k. The unitary irreducible representations are calculated and the carrier space proves to be the set of Shapiro’s wave functions. The generalized equations of motion constitute a simple example of exactly solvable finite-difference set of equations associated with infinite-order polarization equations. Introduction: Higher-order Polarizations One essential difference between Geometric Quantization [1, 2, 3, 4] and a Group Approach to Quantization (GAQ) [5, 6] is the possibility of introducing in the latter higher-order polarizations made out of elements of the (left) enveloping algebra. These higher-order polarizations are especially suitable for those (anomalous) cases in which the

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Renormalization and Statistical Mechanics in Many-Particle Systems. II. Statistical Perturbation Method

Duane C. Wallace

A general nonperturbation problem, described by the Hamiltonian $\mathcal{H}$ , is considered. Approximate creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are defined as operators which satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}+{{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where ${\ensuremath{\omega}}_{i}$ are the creation energies, and the remainder operators ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small in the sense that statistical averages involving the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small. A zeroth approximation is to neglect the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$. In this case, a straight-forward derivation provides general relations between statistical averages such as $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}〉$ and $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}〉$, where $\ensuremath{\Omega}$ is any operator. The usual boson and fermion occupation numbers are a trivial result of these zeroth-order relations. Corrections to the general zeroth-order relations arise from the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$; such corrections are considered in the spirit of perturbation theory. An explicit first-order correction is obtained for the averages $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\theta}}_{i}〉$, and this correction is related to a self-consistent energy-renormalization procedure. The present method is similar to the method of thermodynamic Green's functions in that the derivations given here do not require evaluation of any state vectors or of the partition function. The general results are applied to the Heisenberg ferromagnet problem. At low temperatures, the zeroth-order relations give the Bloch spin-wave results, and an approximate evaluation of the first-order contribution gives the leading term in Dyson's ${T}^{4}$ correction to the spontaneous magnetization. For arbitrary temperatures, the zeroth-order relations give the Tyablikov equations for $〈{S}^{z}〉$ , and Callen's results are obtained from the first-order corrections. These examples illustrate the simplicity of calculating statistical averages of functions of the ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${\ensuremath{\theta}}_{i}$ operators by the present method.A general nonperturbation problem, described by the Hamiltonian $\mathcal{H}$, is considered. Approximate creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are defined as operators which satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}+{{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where ${\ensuremath{\omega}}_{i}$ are the creation energies, and the remainder operators ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small in the sense that statistical averages involving the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small. A zeroth approximation is to neglect the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$. In this case, a straight-forward derivation provides general relations between statistical averages such as $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}〉$ and $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}〉$, where $\ensuremath{\Omega}$ is any operator. The usual boson and fermion occupation numbers are a trivial result of these zeroth-order relations. Corrections to the general zeroth-order relations arise from the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$; such corrections are considered in the spirit of perturbation theory. An explicit first-order correction is obtained for the averages $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\theta}}_{i}〉$, and this correction is related to a self-consistent energy-renormalization procedure. The present method is similar to the method of thermodynamic Green's functions in that the derivations given here do not require evaluation of any state vectors or of the partition function. The general results are applied to the Heisenberg ferromagnet problem. At low temperatures, the zeroth-order relations give the Bloch spin-wave results, and an approximate evaluation of the first-order contribution gives the leading term in Dyson's ${T}^{4}$ correction to the spontaneous magnetization. For arbitrary temperatures, the zeroth-order relations give the Tyablikov equations for $〈{S}^{z}〉$, and Callen's results are obtained from the first-order corrections. These examples illustrate the simplicity of calculating statistical averages of functions of the ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${\ensuremath{\theta}}_{i}$ operators by the present method.

Physical ReviewDec 2, 196621 citations

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)

H(p,q)

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\%.

It has been shown by Wigner that the equations of motion of a quantum-mechanical system with a hamiltonian

H=\frac{1}{2}{p}^{2}+V(q)

need not imply the commutation relation

i[p,q]=1

. Particular examples are furnished by the free particle and the harmonic oscillator. On the other hand, Wigner points out that the above commutation rule does follow from the equations of motion if

V(q)

is of the form

V=aq+b

V=a{q}^{3}

. In the present note, this latter result is generalized to the case of an arbitrary potential

V=a{q}^{n}+b

, where

n

is an odd integer; the last-named potentials are, in fact, particular cases of those satisfying a general criterion contained in the article.

A general nonperturbation problem, described by the Hamiltonian

\mathcal{H}

, is considered. Approximate creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are defined as operators which satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}+{{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where ${\ensuremath{\omega}}_{i}$ are the creation energies, and the remainder operators ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small in the sense that statistical averages involving the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small. A zeroth approximation is to neglect the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$. In this case, a straight-forward derivation provides general relations between statistical averages such as $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}〉$ and $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}〉$, where $\ensuremath{\Omega}$ is any operator. The usual boson and fermion occupation numbers are a trivial result of these zeroth-order relations. Corrections to the general zeroth-order relations arise from the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$; such corrections are considered in the spirit of perturbation theory. An explicit first-order correction is obtained for the averages $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\theta}}_{i}〉$, and this correction is related to a self-consistent energy-renormalization procedure. The present method is similar to the method of thermodynamic Green's functions in that the derivations given here do not require evaluation of any state vectors or of the partition function. The general results are applied to the Heisenberg ferromagnet problem. At low temperatures, the zeroth-order relations give the Bloch spin-wave results, and an approximate evaluation of the first-order contribution gives the leading term in Dyson's

{T}^{4}

correction to the spontaneous magnetization. For arbitrary temperatures, the zeroth-order relations give the Tyablikov equations for

〈{S}^{z}〉

, and Callen's results are obtained from the first-order corrections. These examples illustrate the simplicity of calculating statistical averages of functions of the ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${\ensuremath{\theta}}_{i}$ operators by the present method.