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Journal of Heze University

The Analysis of the Common Eigenstate and Entangled State of Two Particles' Relative Coordination and the Sum of Boson Annihilation Operator

MA Mei-juan

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

Since the commutation relation [x1-x2,a1+a2]=0,their common eigenvector |β,x〉 is constructed by using the technique of integration within an ordered product(IWOP) of operators.This state exhibits the properties of the coherent and entangled states.Its entangled properties and the relations with the EPR entangled state can be confirmed by Schmidt decomposition and the unilateral integral for|β,x〉.The superposition of the states |β=β1+iβ2,x〉 along β1 axis yields the ideal two-mode entangled state.

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Since the commutation relation [x1-x2,a1+a2]=0,their common eigenvector |β,x〉 is constructed by using the technique of integration within an ordered product(IWOP) of operators.This state exhibits the properties of the coherent and entangled states.Its entangled properties and the relations with the EPR entangled state can be confirmed by Schmidt decomposition and the unilateral integral for|β,x〉.The superposition of the states |β=β1+iβ2,x〉 along β1 axis yields the ideal two-mode entangled state.
The new two mode coherent entangled state representation 1p,β〉 is proposed by the technique of integration within an ordered product of operators. Its characteristics and mechanism are analysed. The results show that, the common eigenvector |p,β〉 of p1-p2 and a1+a2 exhibits both the characteristics of the coherent state and entangled state. The 1|p,β〉 makes up a new quantum mechanical representation and possesses the completeness relation. The ideal single-mode squeezed state |p=0〉2=exp［1/2a+22|0〉2 and the vacuum state |0〉1 overlapping on a beam splitter may produce the coherent entangled state |p,β〉at the output by the modulation of the laser field.
The coherent entangled state |β, x〉 is proposed in Fock space, which exhibits both the properties of the coherent and entangled states. The |β, x〉 makes up a new quantum mechanical representation, and the completeness relation of |β, x〉 is proved by virtue of the technique of integral within an ordered product of operators. The corresponding Schmidt decomposition of |β, x〉 is investigated. Furthermore, a feasible experimental scheme of |β, x〉 is presented, and generalized P-representation is constructed in the coherent entangled state |β, x〉.
We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product (IWOP) of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some two-body dynamic problems. Moreover, the pure-state density matrix |ξ 1 , ξ 2 〉 C,D C,D 〈ξ 1 , ξ 2 | is a Radon transform of Wigner operator.