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Science China Physics Mechanics and Astronomy

Coherent state and normal ordering method for transiting Hermite polynomials to Laguerre polynomials

Hong-Yi Fan, Jun Zhou

Published Mar 13, 2012• 12 citations
Publisher
DOI
10.1007/s11433-012-4677-x

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We study the optical field's quadrature excitation state X m |0〉, where X = ( a + a † )/√2 is the quadrature operator. We find it is ascribed to the Hermite-polynomial excitation state. For the first time, we determine this state's normalization constant which turns out to be a Laguerre polynomial. This is due to the integration method within the ordered product of operators (IWOP). The normalization for the two-mode quadrature excitation state is also completed by virtue of the entangled state representation.

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By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials.We also derive the new reciprocal relation of Laguerre polynomials n=0(1) n l n L n(x) = x l l!,and its application in deriving the sum rule of the Wingner function of Fock states is demonstrated.Some new expansion identities about the operator Laguerre polynomial are also derived.This opens a new route of deriving mathematical polynomials formulas by virtute of the quantum mechanical representations and operator ordering technique.
We study the optical field's quadrature excitation state X m |0〉, where X = ( a + a † )/√2 is the quadrature operator. We find it is ascribed to the Hermite-polynomial excitation state. For the first time, we determine this state's normalization constant which turns out to be a Laguerre polynomial. This is due to the integration method within the ordered product of operators (IWOP). The normalization for the two-mode quadrature excitation state is also completed by virtue of the entangled state representation.