# 3: Time-Evolution Operator

• • Contributed by Andrei Tokmakoff
• Professor (Chemistry) at University of Chicago

Dynamical processes in quantum mechanics are described by a Hamiltonian that depends on time. Naturally the question arises how do we deal with a time-dependent Hamiltonian? In principle, the time-dependent Schrödinger equation can be directly integrated choosing a basis set that spans the space of interest. Using a potential energy surface, one can propagate the system forward in small time-steps and follow the evolution of the complex amplitudes in the basis states. In practice even this is impossible for more than a handful of atoms, when you treat all degrees of freedom quantum mechanically. However, the mathematical complexity of solving the time-dependent Schrödinger equation for most molecular systems makes it impossible to obtain exact analytical solutions. We are thus forced to seek numerical solutions based on perturbation or approximation methods that will reduce the complexity. Among these methods, time-dependent perturbation theory is the most widely used approach for calculations in spectroscopy, relaxation, and other rate processes. In this section we will work on classifying approximation methods and work out the details of time-dependent perturbation theory.

• 3.1: Time-Evolution Operator
We are seeking equations of motion for quantum systems that are equivalent to Newton’s—or more accurately Hamilton’s—equations for classical systems. The question is, if we know the wavefunction at a specific time, how does it change with time? How do we determine the wavefunction for some later time? We will use our intuition here, based largely on correspondence to classical mechanics.
• 3.2: Integrating the Schrödinger Equation Directly
Hhow do we evaluate the time-propagator and obtain a time-dependent trajectory for a quantum system? Rather than general recipes, there exist an arsenal of different strategies that are suited to particular types of problems. The choice of how to proceed is generally dictated by the details of your problem, and is often an art-form. Considerable effort needs to be made to formulate the problem, particularly choosing an appropriate basis set for your problem.
• 3.3: Transitions Induced by Time-Dependent Potential
For many time-dependent problems, we can often partition the problem so that the time-dependent Hamiltonian contains a time-independent part (H₀)that we can describe exactly, and a time-dependent potential.  The remaining degrees of freedom are discarded, and then only enter in the sense that they give rise to the interaction potential with H₀. This is effective if you have reason to believe that the external Hamiltonian can be treated classically or if negligible.
• 3.4: Resonant Driving of a Two-Level System
Let’s describe what happens when you drive a two-level system with an oscillating potential. Note, this is the form you would expect for an electromagnetic field interacting with charged particles, i.e. dipole transitions. In a simple sense, the electric field is
• 3.5: Schrödinger and Heisenberg Representations
The mathematical formulation of quantum dynamics that has been presented is not unique. So far, we have described the dynamics by propagating the wavefunction, which encodes probability densities. Ultimately, since we cannot measure a wavefunction, we are interested in observables, which are probability amplitudes associated with Hermitian operators, with time dependence that can be interpreted differently.
• 3.6: Interaction Picture
The interaction picture is a hybrid representation that is useful in solving problems with time-dependent Hamiltonians.
• 3.7: Time-Dependent Perturbation Theory
Perturbation theory refers to calculating the time-dependence of a system by truncating the expansion of the interaction picture time-evolution operator after a certain term. In practice, truncating the full time-propagator U is not effective, and only works well for times short compared to the inverse of the energy splitting between coupled states of your Hamiltonian.
• 3.8: Fermi’s Golden Rule
A number of important relationships in quantum mechanics that describe rate processes come from first-order perturbation theory. These expressions begin with two model problems that we want to work through: (1) time evolution after applying a step perturbation, and (2) time evolution after applying a harmonic perturbation. As before, we will ask: if we prepare the system in one state, what is the probability of observing the system in a different state following the perturbation?