New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

doi:10.1088/1674-1056/19/4/040305

Abstract Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, J. Phys. A 25 (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator $\varDelta \left( q',p'\right) $ ($\mathrm{q}$-number transform) in phase space quantum mechanics, $$\iint_{-\infty}^{\infty}\frac{{\rm d}p'{\rm d}q'}{\pi }\varDelta \left( q',p'\right) {\rm e}^{-2{\rm i}\left( p-p'\right) \left( q-q'\right) }=\delta \left( p-P\right) \delta \left( q-Q\right),$$ and its inverse% $$ \iint_{-\infty}^{\infty}{\rm d}q{\rm d}p\delta \left( p-P\right) \delta \left( q-Q\right) {\rm e}^{2{\rm i}\left( p-p'\right) \left( q-q'\right) }=\varDelta \left( q',p'\right),$$ where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among $Q$--$P$ ordering, $P$--$Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.

Keywords: Wigner operator Weyl ordering two-fold integration transformation

Received: 02 April 2009
Revised: 14 July 2009
Accepted manuscript online:

PACS:	03.65.Vf	(Phases: geometric; dynamic or topological)
	03.65.Fd	(Algebraic methods)
	02.30.Tb	(Operator theory)
	02.30.Uu	(Integral transforms)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos.~10775097 and 10874174), and Specialized Research Fund for the Doctoral Program of Higher Education of China.

Cite this article:

Fan Hong-Yi(范洪义) New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering 2010 Chin. Phys. B 19 040305

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New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

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