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Normal-ordered equivalent of the Weyl ordering of $\hat{q}^j \hat{p}^k$

Hendry M. Lim

Abstract

The problem of quantizing a bivariate dynamical system can be reduced to evaluating the ordering of $\hat{q}^j \hat{p}^k$. Here, we consider the Weyl ordering of $\hat{q}^j \hat{p}^k$ that is then expressed in term of the annihilation $\hat{a}$ and creation $\hat{a}^\dagger$ operator. The explicit formula for the normal-ordered equivalent (all $\hat{a}^\dagger$'s preceeding all $\hat{a}$'s) of the resulting expression is then given, and some relations are discussed.

Normal-ordered equivalent of the Weyl ordering of $\hat{q}^j \hat{p}^k$

Abstract

The problem of quantizing a bivariate dynamical system can be reduced to evaluating the ordering of . Here, we consider the Weyl ordering of that is then expressed in term of the annihilation and creation operator. The explicit formula for the normal-ordered equivalent (all 's preceeding all 's) of the resulting expression is then given, and some relations are discussed.
Paper Structure (9 sections, 37 equations)

This paper contains 9 sections, 37 equations.

Table of Contents

  1. Introduction
  2. Discussion
  3. Quantizing a bivariate dynamical system boils down to ordering qĵ pk̂
  4. The ladder operators
  5. Main result
  6. Symmetry
  7. Concluding Remarks
  8. Obtaining the general form of lambda_klpq and xi_jkpq
  9. Evaluating zeta_jkpq