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.
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.
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′) (q-number transform) in phase space quantum mechanics, ∬−∞∞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′), where Q,P are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas 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.
Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ω k ( p , q ) with a real k parameter and can unify the P—Q , Q—P , and Weyl ordering of operators in k = 1, − 1, 0, respectively, we find the mutual transformations between δ ( p — P ) δ ( q — Q ), δ ( q — Q ) δ ( p — P ), and Ω k ( p, q ), which are, respectively, the integration kernels of the P—Q , Q—P , and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The — and — ordered forms of Ω k ( p , q ) are also derived, which helps us to put the operators into their — and — ordering, respectively.
Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator ?k(p,q) with a real k parameter and can unify the P–Q, Q–P, and Weyl ordering of operators in k = 1,-1, 0, respectively, we find the mutual transformations between δ(p- P)δ(q- Q), δ(q- Q)δ(p- P), and Ωk(p,q), which are, respectively, the integration kernels of the P–Q, Q–P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The P- and Q- ordered forms of Ωk(p,q) are also derived,which helps us to put the operators into their P- and Q- ordering, respectively.
Since the foundation of quantum mechanics, operator-ordering identities for mutual transformation of power of coordinate-momentum operators have been a fundamental and tough topic. To the best of our knowledge, this topic has not been tackled smoothly because there is no elegant and direct way to investigate it. In this paper we report a very concise and novel method to handle this topic, i.e., we employ the generating function of two-variable Hermite polynomial and the characteristics of ordered operators to derive a series of operator-ordering identities for mutual transformation of power of coordinate-momentum operators: they surly possess potential applications. The essence of our method lies in the fact that coordinate-momentum operators can be permutable within ordered product of operators, just as the scenarios in P-Q ordering, Q-P ordering and Weyl ordering. We also derive the integration transformation formula about two-variable Hermite polynomial in phase space. The correspondence relation between operator ordering and quantization recipe is established. The beauty of theoretical physics is embodied extensively in the paper.
Starting from the optical fractional Fourier transform (FFT) and using the technique of integration within an ordered product of operators we establish a formalism of FFT for quantum mechanical wave functions. In doing so, the essence of FFT can be seen more clearly, and the FFT of some wave functions can be derived more directly and concisely. We also point out that different FFT integral kernels correspond to different quantum mechanical representations. They are generalized FFT. The relationship between the FFT and the rotated Wigner operator is studied by virtue of the Weyl ordered form of the Wigner operator.