Adiabatic Theorem - Deriving Conditions For Diabatic Vs Adiabatic Passage

Deriving Conditions For Diabatic Vs Adiabatic Passage

We will now pursue a more rigorous analysis. Making use of bra-ket notation, the state vector of the system at time can be written

,

where the spatial wavefunction alluded to earlier is the projection of the state vector onto the eigenstates of the position operator

.

It is instructive to examine the limiting cases, in which is very large (adiabatic, or gradual change) and very small (diabatic, or sudden change).

Consider a system Hamiltonian undergoing continuous change from an initial value, at time, to a final value, at time, where . The evolution of the system can be described in the Schrödinger picture by the time-evolution operator, defined by the integral equation

,

which is equivalent to the Schrödinger equation.

,

along with the initial condition . Given knowledge of the system wave function at, the evolution of the system up to a later time can be obtained using

The problem of determining the adiabaticity of a given process is equivalent to establishing the dependence of on .

To determine the validity of the adiabatic approximation for a given process, one can calculate the probability of finding the system in a state other than that in which it started. Using bra-ket notation and using the definition, we have:

.

We can expand

.

In the perturbative limit we can take just the first two terms and substitute them into our equation for, recognizing that

is the system Hamiltonian, averaged over the interval, we have:

.

After expanding the products and making the appropriate cancellations, we are left with:

,

giving

,

where is the root mean square deviation of the system Hamiltonian averaged over the interval of interest.

The sudden approximation is valid when (the probability of finding the system in a state other than that in which is started approaches zero), thus the validity condition is given by

,

which is a statement of the time-energy form of the Heisenberg uncertainty principle.