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Communications in Theoretical Physics

Coordinate-Momentum Intermediate Representation and Marginal Distributions of Quantum Mechanical Bivariate Normal Distribution

Fan Hong-Yi, Lou Sen‐Yue

Published Mar 1, 2008• 1 citation
Publisher

Abstract

We introduce bivariate normal distribution operator for state vector |ψ〉 and find that its marginal distribution leads to one-dimensional normal distribution corresponding to the measurement probability | λ,ν 〈x|·ψ〉| 2 , where |x〉 λ,ν is the coordinate-momentum intermediate representation. As a by-product, the one-dimensional normal distribution in statistics can be explained as a Radon transform of two-dimensional Gaussian function.

DOI
10.1088/0253-6102/49/3/18

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We introduce bivariate normal distribution operator for state vector |ψ〉 and find that its marginal distribution leads to one-dimensional normal distribution corresponding to the measurement probability | λ,ν 〈x|·ψ〉| 2 , where |x〉 λ,ν is the coordinate-momentum intermediate representation. As a by-product, the one-dimensional normal distribution in statistics can be explained as a Radon transform of two-dimensional Gaussian function.
The state |x〉λ,v was derived by using the technique of integration within an ordered product of operators(IWOP)that was the eigenstate of two rotational quadrature components λQ^+vP of light field.The basic vector |x〉λ,v becomes a new quantum mechanics representation which is between coordinate representation and momentum representation.With the help of the new representation,we obtain the normal product of the operators exp[-f(λQ +vP )2] and(λQ+vP)n.Furthermore,we construct a new quantum unitary operator which is used on vacuum state can obtain a new quantum optics squeezing state.
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).
Based on the technique of integration within an ordered product of operators we investigate a completeness relation of pure states (such as the coordinate eigenstate, the momentum eigenstate and the coherent state) into normally ordered Gaussian forms. The Weyl ordering invariance under similarity transformations is employed to reveal physical meaning of a kind of normally ordered Gaussian operators, which have the similar forms to the bivariate normal distributions in statistics, i.e., the thermo mixed state density matrix.