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Quantum Dynamics Tool Kit

 

 The propagator is a realization of the evolution operator a mapping step which carries the wavefunction from time t  to t':         \(\Psi(t)={\bf U}(t) \Psi(0)\)

The stage of events is the time-energy phase space. The time dependent Schrodinger equation:

\(i \hbar \frac{\partial}{\partial t} \Psi = {\bf H} \Psi\)

A formal solution is obtained by:

\(\Psi(t) = e^{-\frac{i}{\hbar} {\bf H} t} \Psi(0)\)

where \({\bf U}(t) = e^{-\frac{i}{\hbar}{\bf H}t}\)

with the equation of motion

\(i \hbar \frac{\partial}{\partial t}{\bf U} = {\bf H} {\bf U}\)

Our numerical approach is to expand the evolution operator in a polynomial in H. [1,2].

For time independent hamiltonian operators we found that the best polynomial expansion is based on the Chybychev expansion.

A more difficult problem is when we have explicit rime dependence in the Hamiltonian either directly or through  nonlinearity: \({\bf H } \equiv {\bf H}(\psi,t)\). 

This results in a nonlinear equation of motion. In addition, we include an inhomogeneous source term \(s(t)\). The time-dependent nonlinear inhomogeneous Schr¨odinger equation reads:

\(i \hbar \frac{\partial}{\partial t} \Psi = {\bf H}(t,\Psi) \Psi+s(t)\)

If we can split the Hamiltonis to \({\bf H}= {\bf H}_0 + {\bf V}(\Psi(t),t)\)Formaly we have the solution:  \(\Psi (t) = U_0(t)\Psi(0) + \int_0^t U_0(t-\tau){\bf V}(\Psi(t),t)\Psi(\tau)d\tau \)where:  \(U_0(t) = e^{-i H_0 t}\)
[1]H. Tal Ezer and R. Kosloff, An Accurate and Efficient Scheme for Propagating the Time Dependent Schrödinger Equation, J. Chem. Phys., 81, 3967-3970 (1984). 
 
[2] R. Kosloff, Propagation Methods for Quantum Molecular Dynamics, Annu. Rev. Phys. Chem., 45, 145-178 (1994).