Generalized Positive-Definite Operator in Quantum Phase Space Obtained by Virtue of the Weyl Quantization Rule

We introduce a generalized positive-definite operator Δ_g(q,p) by smoothing out the Wigner operator Δ_w(q,p) and by averaging over the "coarse graining'' function. The function is then regarded as the classical Weyl correspondence of the operator Δ_g(q,p); in this way we can easily identify a quantum state |Φ> such that Δ_g(q,p)=|Φ><Φ|, and |Φ> turns out to be a new kind of squeezed coherent state. Correspondingly, the generalized distribution function for any state |φ> is <φ| Δ_g(q,p) |φ> =|<Φ|φ>|² , which is obviously positive-definite and is a generalization of the Husimi function.