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Journal of Mathematical Physics

Eigenkets of the q-deformed creation operator

Ye‐Jun Xu, Jun Song, Hong-Yi Fan, Qiuyu Liu

Published Sep 1, 2010• 2 citations
Publisher

Abstract

By using the contour integral representation of δ-function and the technique of integration within an ordered product of operators, we point out that the q-deformed creation operator possesses the eigenkets. A set of new completeness and orthogonality relations composed of the kets and bras which are not mutually Hermitian conjugates are derived. Application of the completeness relation in constructing the generalized P-representation of density operator is demonstrated.

DOI
10.1063/1.3485041

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By using the contour integral representation of δ-function and the technique of integration within an ordered product of operators, we point out that the q-deformed creation operator possesses the eigenkets. A set of new completeness and orthogonality relations composed of the kets and bras which are not mutually Hermitian conjugates are derived. Application of the completeness relation in constructing the generalized P-representation of density operator is demonstrated.
The q-deformed entangled states are introduced by using deformation quantization methods and new normal ordering of the vacuum projection operator for q-deformed boson oscillator. In similar way, by virtue of the technique of integration within an ordered product (IWOP) of operators, the new completeness and orthogonality relations composed of the bra and ket, which are not mutually Hermitian conjugates are obtained. Furthermore, the property of squeezing operator represented by the q-deformed entangled states is exhibited. Lastly, the nonclassical properties of the q-deformed two-mode squeezed vacuum state are studied.
THE EIGENVECTORS OF q-DEFORMED CREATION OPERATOR aq+a_q^+aq+​ AND THEIR PROPERTIES
Using the completeness relation composed of the coherent state and of the eigenket of bosonic creation operator, we establish a one-to-one correspondence between the z -transform and the quantum-mechanical transform from the representation by number states to the representation by coherent states (Bargmann representation). In this way, the quantum-mechanical version of the various properties of z -transform are obtained and the operators for embodying these properties in the Fock space are derived, which may find applications in quantum states engineering.
Using the resolution of unity composed of bosonic creation operator's eigenkets and annihilation operator's un-normalized eigenket, which is a new quantum mechanical representation in contour integration form, we derive new contour integration expression of associated Laguerre polynomials L ρ m (| z | 2 ) and its generalized generating function formula. A series of recursive relations regarding to L ρ m (| z | 2 ) are also deduced in the context of the Fock representation by algebraic method.