Squeezing and quantum canonical transformations (Q1597514)

An approach to the time-dependent quantum-mechanical canonical transformations (TDCT) is proposed which is based upon group theory and makes use of squeezing operators. TDCT is defined as a unitary transformation \(W(t)\) transforming a given Hamiltonian \(H(t)\) into \(\tilde H(t) = WH(t)W^\dag - i\hbar W\dot W^\dag\). The main result is that the operators \(W(t)\) are connected with ``squeezing operators'' obtained by exponentiating the generators \(K_{0,\pm}\) of the \(\text{SU}(1,1)\) group. In the simplest case the generators \(K_{0,\pm}\) are quadratic expressions of the creation-annihilation operators \(a, a^\dag\) constructed in the usual way of the coordinates and momenta. The set of TDCT \(W(t)\) as well as the set of the squeezing operators form a group.