A DERIVATION OF THE WEYL ORDERED FORM OF TWO-MODE FRESNEL OPERATOR BY VIRTUE OF THE WEYL OPERATOR ORDERING METHOD

Based on the technique of integration with a Weyl ordered product (IWWOP) for the two-mode operator, we derive out the Weyl ordered form of two-mode Fresnel operator (TFO). The multiplication rule for TFO and the matrix element of Weyl ordered form of TFO in coordinate eigenstates are also discussed.

FRESNEL OPERATOR, SQUEEZED STATE AND WIGNER FUNCTION FOR CALDIROLA–KANAI HAMILTONIAN

Based on the technique of integration within an ordered product (IWOP) of operators, we introduce the Fresnel operator for converting a kind of time-dependent Hamiltonian into the standard harmonic oscillator Hamiltonian. The Fresnel operator with the parameters A, B, C, D corresponds to classical optical Fresnel transformation, these parameters are the solution to a set of coupled partial differential equations set up in the above-mentioned converting process. In this way, the Caldirola–Kanai Hamiltonian has been easily converted into the standard harmonic oscillator Hamiltonian. And then the exact wave function solution of the Schrödinger equation governed by the Caldirola–Kanai Hamiltonian is obtained, which represents a squeezed number state. The corresponding Wigner function is derived by virtue of the Weyl ordered form of the Wigner operator and the order-invariance of Weyl ordered operators under similar transformations.