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Chinese Physics Letters

Two-Parameter Radon Transformation of the Wigner Operator and Its Inverse

Fan Hong‐Yi, Hailing Cheng

Published Jul 1, 2001• 23 citations
Publisher

Abstract

Using the technique of integration within an ordered product of operators, we reveal that a new quantum mechanical representation |x,µ,ν exists, the eigenvector of operator µQ + νP (linear combination of coordinate Q and momentum P) with eigenvalue x, which is inherent to the two-parameter (µ,ν) Radon transformation of the Wigner operator. It turns out that the projection operator |x,µ,νx,µ,ν| is just the Radon transformation of the Wigner operator. The inverse of operator Radon transformation is also derived which indicates tomography in operator version.

DOI
10.1088/0256-307x/18/7/303

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Using the technique of integration within an ordered product of operators, we reveal that a new quantum mechanical representation |x,µ,ν exists, the eigenvector of operator µQ + νP (linear combination of coordinate Q and momentum P) with eigenvalue x, which is inherent to the two-parameter (µ,ν) Radon transformation of the Wigner operator. It turns out that the projection operator |x,µ,νx,µ,ν| is just the Radon transformation of the Wigner operator. The inverse of operator Radon transformation is also derived which indicates tomography in operator version.
Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector |q⟩ λ,ν of linear combination of the coordinate and momentum operator, (λQ + νP), where λ, ν are real numbers, and |q⟩ λν is complete, then the projector |q⟩ λν ⟨q| must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.
By searching for Q- and P-ordering form of the coordinate-momentum intermediate representation |x⟩λ,νλ,ν⟨x| (formed by the eigenvector of λQ+νP with eigenvalue being x, Q,P=iħ), and noting that |x⟩λ,νλ,ν⟨x|=δ(x-λQ-νP) is just the Radon transformation of the Wigner operator, we find new relationship between a quantum state ψ’s tomogram and its wave function in coordinate (or momentum) representation, i.e. the tomogram can be split into two parts, one is ψq’s Gaussian integration transformation with the parameter ν and another is ψ∗p’s Gaussian integration transformation with the parameter λ. In this way, the quantum tomogram of number state is conveniently deduced. We also derive Radon transform of δ(p-P)δ(q-Q) and δ(q-Q)δ(p-P), which can be either viewed as P- and Q-ordered operator correspondence of the classical function δ(x-λq-νp).
We show that for tomographic approach there exist two mutual conjugate quantum states | p ,σ,τ⟩ and | x ,μ,ν⟩ (named the intermediate coordinate-momentum representation), and the two Radon transforms of the Wigner operator are just the pure-state density matrices | p ⟩ σ,τσ,τ ⟨ p | and | x ⟩ λ,νλ,ν ⟨ x | respectively. As a result, the tomogram of quantum states can be considered as the module-square of the states' wave function in these two representations. Throughout the paper we fully employ the technique of integration within an ordered product of operators. In this way we establish a new convenient formalism of quantum tomogram.
The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the latter is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and non-normalizable eigenfunctions to the number operator is considered from the point of view of this new representation.