Quantum Mechanics in Fock Space

In the present paper we propose to develop a quantum-mechanical scheme in Fock space that would describe interactions that take place through the formation of a compound particle. The discussion will be restricted to a dynamical system representing a single-level scattering process. The state of this dynamical system can be found in two stages: initial particles and compound nucleus. The dynamical behavior of this state will be given in terms of a hamiltonian (that has no classical analog) which will be derived from the boundary conditions satisfied by the state. The representation of the dynamical variables, associated with the two stages of the state, will be discussed, and a complete set of constants of motion of our dynamical system will be given. The expansion of an arbitarary Fock state in terms of the simultaneous eigenstates of this complete set of constants of motion, leads to the generalized Hankel transforms, used in a preceding article for the determination of the time dependent states for scattering and disintegration. In a representation in which the constants of motion are diagonal, the operator representing the relative kinetic energy of the two initial particles will be nondiagonal, particularly in the neighborhood of the resonance energy. The unitary matrix connecting any initial Fock state with the corresponding state at time $t$ will be obtained, and with its help the time-dependent behavior of the Heisenberg dynamical variables of our dynamical system can be derived.