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arXiv:math-ph/0512021 · Modern Physics Letters A

SIMULTANEOUS EIGENSTATES OF THE NUMBER-DIFFERENCE OPERATOR AND A BILINEAR INTERACTION HAMILTONIAN DERIVED BY SOLVING A COMPLEX DIFFERENTIAL EQUATION

HONG-YI FAN, WEI-BO GAO

Published Dec 14, 2006• 2 citations
arXivHTMLPDF

Abstract

As a continuum work of Bhaumik et al.3 who derived the common eigenvector of the number-difference operator Q≡(a†a-b†b) and pair-annihilation operator ab, we search for the simultaneous eigenvector of Q and (ab-a†b†) by setting up a complex differential equation in the bipartite entangled state representation. The differential equation is then solved in terms of the two-variable Hermite polynomials and the formal hypergeometric functions. The work is also an addendum to Ref. 5 in which the common eigenkets of Q and pair creators a†b† are discussed.

DOI
10.1142/s0217732306020275

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As a continuum work of Bhaumik et al.3 who derived the common eigenvector of the number-difference operator Q≡(a†a-b†b) and pair-annihilation operator ab, we search for the simultaneous eigenvector of Q and (ab-a†b†) by setting up a complex differential equation in the bipartite entangled state representation. The differential equation is then solved in terms of the two-variable Hermite polynomials and the formal hypergeometric functions. The work is also an addendum to Ref. 5 in which the common eigenkets of Q and pair creators a†b† are discussed.
For complex scalar fields ( x ) and ( x ), we introduce the entangled states which are their common eigenvectors. The states possessing complete and orthonormal properties are a field-theoretical generalization of the Einstein-Podolsky-Rosen pair states in quantum mechanics. Based on the new representation, we obtain the complete and orthonormal states { q }, r , the common eigenvectors of the charge operator and ( x ) ( x ) , with which the charge lowering and raising operators can be naturally defined. The applications of our formulation are briefly discussed.