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.
Read more about this topic: Adiabatic Theorem
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