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Chinese Physics B

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

Fan Hong-Yi

Published Apr 1, 2010• 6 citations
Publisher

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 Δ ( q',p' ) (q-number transform) in phase space quantum mechanics, and its inverse 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.

DOI
10.1088/1674-1056/19/4/040305

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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 Δ ( q',p' ) (q-number transform) in phase space quantum mechanics, and its inverse 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.
Using operators' Weyl ordering expansion formula (Hong-yi Fan,\emph{\}J. Phys. A 25 (1992) 3443) we find new two-fold integration transformation about the Wigner operator Δ(q′,p′)Δ(q',p')Δ(q′,p′) (qqq-number transform) in phase space quantum mechanics, ∬−∞∞dp′dq′/πΔ(q′,p′)e−2i(p−p′)(q−q′)=δ(p−P)δ(q−Q), \iint_{-\infty}^\infty dp' dq'/πΔ(q',p') e^{-2i(p-p') (q-q')} =δ(p-P) δ(q-Q), ∬−∞∞​dp′dq′/πΔ(q′,p′)e−2i(p−p′)(q−q′)=δ(p−P)δ(q−Q), and its inverse ∬−∞∞dqdpδ(p−P)δ(q−Q)e2i(p−p′)(q−q′)=Δ(q′,p′),\iint_{-\infty}^\infty dq dp δ(p-P) δ(q-Q) e^{2i(p-p') (q-q')}=Δ(q',p'), ∬−∞∞​dqdpδ(p−P)δ(q−Q)e2i(p−p′)(q−q′)=Δ(q′,p′), where Q,Q,Q, PPP are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas among Q−PQ-PQ−P ordering, P−QP-QP−Q ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched.
As a natural and important extension of Fan's paper (Fan Hong-Yi 2010 Chin. Phys. B 19 040305) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator Δ (μ, ν) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to ( a † 1 – a 2 ) and ( a † 1 – a 2 ) can be solved and the contents of phase space quantum mechanics can be enriched, where a i , a † i are bosonic creation and annihilation operators, respectively.
In this paper, it can be found that there is a type of integra-transformation which corresponds to a quantum mechanical fundamental commutative relation, with its integral kernel being 1/exp[2i(q-Q)(p-P)], here denotes Weyl ordering, and Q and P are the coordinate and the momentum operator, respectively. Such a transformation is responsible for the mutual-converting among three ordering rules(P-Q ordering, Q-P ordering and Weyl ordering). We also deduce the relationship between this kernel and the Wigner operator, and in this way a new approach for deriving Wigner function in quantum states is obtained.
Based on the two mutually conjugate entangled state representations |ξ⟩ and |η⟩, we propose an integration transformation in ξ — η phase space , and its inverse transformation, which possesses some well-behaved transformation properties, such as being invertible and the Parseval theorem. This integral transformation is a convolution, where one of the factors is fixed as a special normalized exponential function. We generalize this transformation to a quantum mechanical case and apply it to studying the Weyl ordering of bipartite operators, regarding to ( Q 1 – Q 2 ) ↔ ( P 1 – P 2 ) ordered and simultaneously ( P 1 + P 2 ) ↔ ( Q 1 + Q 2 ) ordered operators.