It is shown that the knowledge of the solution of the Heisenberg equations for a given Hamiltonian allows us to find the corresponding propagator up to a time-dependent phase factor, which gives the solution of the Schrödinger equation.
In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).
We find the form of the potential depending on the coordinates and the time such that a solution, S, of the Hamilton--Jacobi equation yields an exact solution, exp(iS/ℏ), of the corresponding Schrödinger equation.
Numerical calculations on two-dimensional spin-1/2 XXZ Heisenberg models are presented. The computational technique entails solving the time-dependent Schr\"odinger equation. The time propagation of wave functions is performed employing Trotter-Suzuki product formulas. The algorithms are numerically stable and are used to study models involving \ensuremath{\approxeq}107 states. With this approach both the thermodynamic properties as well as the many-particle dynamics can be obtained. Densities of states and zero-temperature dynamic magnetic structure factors are presented. For the isotropic Heisenberg antiferromagnet thermodynamic properties are computed for lattices up to 26 sites.