Optical field’s quadrature excitation studied by new Hermite-polynomial operator identity

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.