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arXiv:1211.4877 · Journal of Physics A Mathematical and Theoretical

Commutation relations of operator monomials

Jean‐Christophe Pain

Published Dec 18, 2012• 10 citations
arXivHTMLPDF

Abstract

In this short paper, the commutator of monomials of operators obeying constant commutation relations is expressed in terms of anti-commutators. The formula involves Bernoulli numbers or Euler polynomials evaluated in zero. The role of Bernoulli numbers in quantum-mechanical identities such as the Baker–Campbell–Hausdorff formula is emphasized and applications connected to ordering problems as well as to the Ehrenfest theorem are proposed.

DOI
10.1088/1751-8113/46/3/035304

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In this short paper, the commutator of monomials of operators obeying constant commutation relations is expressed in terms of anti-commutators. The formula involves Bernoulli numbers or Euler polynomials evaluated in zero. The role of Bernoulli numbers in quantum-mechanical identities such as the Baker–Campbell–Hausdorff formula is emphasized and applications connected to ordering problems as well as to the Ehrenfest theorem are proposed.
Abstract The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators X and Y, according to the Campbell-Baker-Hausdorff-Dynkin theorem, e X+Y is not equivalent to e X e Y , but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators {X, Y, Z}, such that [X, Y] = κ Z for scalar κ and cyclic permutations, here it is proven that e a X+b Y is equivalent to e p Z e q X e−p Z for scalar p and q. Extensions for e a X+b Y+c Z are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields.
As is well known, the binomial theorem is a classical mathematical relation that can be straightforwardly proved by induction or through a Taylor expansion, albeit it remains valid as long as [A,B]=0. In order to generalize such an important equation to cases where [A,B]≠0, an algebraic approach based on Cauchy’s integral theorem in conjunction with the Baker–Campbell–Hausdorff series is presented that allows a partial extension of the binomial theorem when the commutator [A,B]=c, where c is a constant. Some useful applications of the new proposed generalized binomial formula, such as energy eigenvalues and matrix elements of power, exponential, Gaussian, and arbitrary f(x̂) functions in the one-dimensional harmonic oscillator representation are given. The results here obtained prove to be consistent in comparison to other analytical methods.
The problem of ordering operators has afflicted quantum mechanics since its foundation. Several orderings have been devised, but a systematic procedure to move from one ordering to another is still missing. The importance of establishing relations among different orderings is demonstrated by Wick's theorem (which relates time ordering to normal ordering), which played a crucial role in the development of quantum field theory. We prove the general ordering theorem (GOT), which establishes a relation among any pair of orderings, that act on operators satisfying generic (i.e., operatorial) commutation relations. We expose the working principles of the GOT by simple examples, and we demonstrate its potential by recovering two famous algebraic theorems as special instances: the Magnus expansion and the Baker-Campbell-Hausdorff formula. Remarkably, the GOT establishes a formal relation between these two theorems, and it provides compact expressions for them, unlike the notoriously complicated ones currently known.
Quantization under simplest symmetric ordering of classical monomials with negative integral exponents of momentum has been written in integral form. This integral representation not only proved useful in transforming Weyl quantization to symmetric ordering quantization, and vice versa. They also facilitate derivation of commutation relations among different operators involving inverse powers of momentum operator, relations which can be used in solving quantization problems for classical functions with singularity along the momentum axis.
We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.