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Journal of Anhui University

The common eigenstate of coordinate and momentum crossing compounding operator and its properties

Gang Ren

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

In this paper,the common eigenstate of coordinate and momentum crossing compounding with parameter |η〉was derived by using the technique of intergation within an ordered product of operators(IWOP).Afterwards,we proved that |η〉was orthogonal and entanglement.So we could say that the |η〉was a new kind of representation.On the basis,we easily obtained the normal product of the operator identities in the ordering form with the help of the new representation and IWOP technique.It was clear that the new representation had its advantages in this aspect of problem solution.

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In this paper,the common eigenstate of coordinate and momentum crossing compounding with parameter |η〉was derived by using the technique of intergation within an ordered product of operators(IWOP).Afterwards,we proved that |η〉was orthogonal and entanglement.So we could say that the |η〉was a new kind of representation.On the basis,we easily obtained the normal product of the operator identities in the ordering form with the help of the new representation and IWOP technique.It was clear that the new representation had its advantages in this aspect of problem solution.
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.
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.
By using the parameter differential method of operators, we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings, which is more ecumenical, simpler, and neater than the existing ways. These products are very useful in obtaining some new differential relations and useful mathematical integral formulas. Further, we derive the normally ordered form of the operator (fQ + gP)−n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate–momentum representation. In addition, general mutual transformation rules of the normal and anti-normal orderings, which have good universality, are derived and hence the anti-normal ordering of (fQ + gP)−n is also obtained. Finally, the application of some new identities is given.