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College Physics

Research on quantum mechanical representations via the two-dimensional normal distribution function in statistics

Fan Hong-Yi

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Based on the two-dimensional normal distributions in statistics and the technique of integration within an ordered product of operators(IWOP),the explicit forms in the Fock space of some quantum representations are simply obtained such as coordinate representation,momentum representation,coordinate-momentum intermediate representation,and coherent representation.In addition,their corresponding completeness is also easily proved.

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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.

Communications in Theoretical Physics•Mar 1, 2008•1 citation

Ridgelet transform research based on the coordinate representation

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Inspired by the wavelet transform in quantum mechanics, we define the new Ridgelet transform for quantum optics by rewriting the classic Ridgelet transform via the two-mode coordinate representation in Fock space. Furthermore, we give the explicit form of the asymmetric operator's integral and derive two useful formulas for the normal ordering of the two-mode operator with the help of the technique of integration within an ordered product (IWOP) of operators. By choosing the two-variable Mexican hat's mother wavelet function, we analyse the Ridgelet transforms of the coherent state, special squeezed coherent state, intermediary entangled state on the basis of the theories we have mentioned.

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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.

Jilin Normal University Journal•Jan 1, 2011•0 citations

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In this paper by virtue of the technique of integration within an ordered product (IWOP) of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator ( fQ + gP ) n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.

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By finding that the projection operators of the basic quantum mechanical representations can be written as operator delta functions, we derive the explicit forms of quantum-mechanical fundamental representations in Fock space by directly carrying out the integrals within the normally ordered product of operators.

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Physics of a Kind of Normally Ordered Gaussian Operators in Quantum Optics

Fan Hong‐Yi, Hongqi Li

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.

Chinese Physics Letters•Dec 1, 2007•9 citations
Based on the two-dimensional normal distributions in statistics and the technique of integration within an ordered product of operators(IWOP),the explicit forms in the Fock space of some quantum representations are simply obtained such as coordinate representation,momentum representation,coordinate-momentum intermediate representation,and coherent representation.In addition,their corresponding completeness is also easily proved.
Inspired by the wavelet transform in quantum mechanics, we define the new Ridgelet transform for quantum optics by rewriting the classic Ridgelet transform via the two-mode coordinate representation in Fock space. Furthermore, we give the explicit form of the asymmetric operator's integral and derive two useful formulas for the normal ordering of the two-mode operator with the help of the technique of integration within an ordered product (IWOP) of operators. By choosing the two-variable Mexican hat's mother wavelet function, we analyse the Ridgelet transforms of the coherent state, special squeezed coherent state, intermediary entangled state on the basis of the theories we have mentioned.
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.
In this paper by virtue of the technique of integration within an ordered product (IWOP) of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator ( fQ + gP ) n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.
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.