5.1: Adiabatic Processes

An adiabatic process is one that occurs so slowly that a system’s internal degrees of freedom can fully and continuously adapt to externally driven changes. In quantum mechanics, the adiabatic approximation refers to those solutions to the Schrödinger equation that use a time-scale separation between fast and slow degrees of freedom to assert that slowly moving particles are effectively static on the timescale of motion by fast particles. Therefore, slow particles evolve under the influence of fully equilibrated fast particles. The adiabatic approximation allows one to find approximate wavefunctions that are product states in the fast and slow degrees of freedom.

Perhaps the most fundamental and commonly used version is the Born-Oppenheimer Approximation (BOA), which underlies much of how we conceive of molecular electronic structure and is the basis of potential energy surfaces. The BOA assumes that the motion of electrons is much faster than nuclei due to their large difference in mass, and therefore electrons adapt very rapidly to any changes in nuclear geometry. That is, the electrons "adiabatically follow" the nuclei. As a result, we can solve for the electronic state of a molecule for fixed nuclear configurations. Gradually stepping nuclear configurations and solving for the energy leads to an adiabatic potential energy surface. Much of our descriptions of chemical reaction dynamics is presented in terms of propagation on these potential energy surfaces. The barriers on these surfaces are how we describe the rates of chemical reactions and transition state. The trajectories along these surfaces are used to describe mechanism.

More generally, the adiabatic approximation is applied in other contexts in which there is a time-scale separation between fast and slow degrees of freedom. For instance, in the study of vibrational dynamics when the bond vibrations of molecules occur much faster than the intermolecular motions of a liquid or solid. This assumption is inherent in classical molecular dynamics force fields with fixed bond lengths. It is also generally implicit in a separation of the Hamiltonian into a system and a bath, a method we will often use to solve condensed matter problems. As widely used as the adiabatic approximation is, there are times when it breaks down, and it is important to understand when this approximation is valid, and the consequences of when it is not. This will be particularly important for describing time-dependent quantum mechanical processes involving transitions between potential energy sources.

In the context of quantum mechanics, adiabatic dynamics refers to time-dependent changes to the Hamiltonian from an initial state \(H_{i}(t_{0})\) to a final form \(H_{f}(t_{f})\) that occur slowly enough that a particle initially in the \(n^{\text {th }}\) eigenstate remains in that eigenstate throughout the process. We have already encountered this. For instance, we saw in the Example in Section 3.6 that when we squeeze and release the potential of a harmonic oscillator on a time scale that is much longer that the inverse of its frequency ( \(\sigma \Omega \gg 1\) ), that a system prepared on the ground state does not transfer any population to other eigenstates. The wavefunction adapts to the stiffening potential and then returns to its original state as the compression is released. Here the adiabatic time-scale separation is between the slow application of the external potential and the fast oscillation period of the oscillator.

An adiabatic process is therefore fully reversible. That is, after propagating the system forward under our time-dependent Hamiltonian from initial state \(\psi_{i}(t_{i}=0)\) to the final state \(\psi_{f}(t_{f})\), we can reverse the dynamics by switching the sign of the Hamiltonian’s time-arguments, and propagate forward in time, but return to the initial state unchanged: \(\psi_{i}(2 t_{f})\). More formally, the adiabatic approximation means that the Hamiltonian commutes with itself for all times.

In practice, this means that we can compute adiabatic dynamics with a time-dependent Hamiltonian by taking small time steps and solve the time-independent Schrödinger equation at each step to obtain slowly evolving eigenstates and energy eigenvalues.

\[H(t) \psi_{n}(t)=E_{n}(t) \psi_{n}(t)\]

As we discussed in Section XX, the time-propagator in the adiabatic picture is expressed in the time-evolving energy eigenvalues

\[\hat{U}(t, t_{0})=\sum_{n}|n\rangle\langle n| \exp \left[-\frac{i}{\hbar} \int_{t_{0}}^{t} E_{n}(t^{\prime}) d t^{\prime}\right]\]

The eigenstate amplitudes of a superposition do not change, but only acquire a time-dependent phase

\[\left|\Psi (t, t_{0} ) \right\rangle = \sum_{n} c_{n} \exp \left[-\frac{i}{\hbar} \int_{t_{0}}^{t} E_{n} (t^{\prime} ) d t^{\prime}\right]|n\rangle\]

Adiabatic dynamics such as this are widely used, for instance in semiclassical methods, empirical valence bond dynamics, and ab initio molecular dynamics.

---

References

\(\tag {1} \label {Cite1}\)

Tannor, D. J., Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Books: Sausilito, CA, 2007; Ch. 9 and Ch. 12.