Quantum Canonical Transformations and Exact Solution of the Schreodinger Equation

Abstract

Time-dependent unitary transformations are used to study the Schreodinger equation for explicitly time-dependent Hamiltonians of the form $H(t)=\vec R(t).\vec J$, where $\vec R$ is an arbitrary real vector-valued function of time and $\vec J$ is the angular momentum operator. The solution of the Schreodinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case $\vec R=(1,0,ν(t))$. This corresponds to the problem of driven two-level atom for the spin half representation of $\vec J$. It is also shown that by requiring the magnitude of $\vec R$ to depend on its direction in a particular way, one can solve the Schreodinger equation exactly. In particular, it is shown that for every Hamiltonian of the form $H(t)=\vec R(t)\cdot \vec J$ there is another Hamiltonian with the same eigenstates for which the Schreodinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holonomy calculation for SU(2) principal bundles (Yang-Mills gauge theory) is also pointed out.