Eigenfunctions of quadratic Hamiltonians in the Wigner representation

Exact solutions of the Schrödinger equation in the Wigner representation are obtained for an arbitrary time-dependent $N$-dimensional quadratic Hamiltonian. It is shown that a complete system of solutions can always be chosen in the form of products of $N$ Laguerre polynomials having arguments that are quadratic integrals of the motion of the corresponding classical problem. The generating function found for the transition probabilities between the Foek states is a multidimensional generalization of Husimi's well-known expression for an oscillator with variable frequency. The motion of a charged particle in a uniform time-dependent electromagnetic field is considered in detail as an example.