The adiabatic approximation
In time independent perturbation theory, we saw that a time dependent perturbation can cause transitions
For this to occur, frequency of the perturbation had to match the spacing between those levels.
But what if
This is a very slow perturbation, which is what we call the adiabatic limit. We expect system initially in eigen state will stay in that state (no transitions), but eigenstate itself will slowly change in time.
Example. Harmonic oscillator with time dependence .
Suppose system starts off in ground state at time .
If spring constant changes very slowly, we expect the only effect is that becomes time dependent
If the time dependence is slow, the solution of the time dependent Schrodinger equation
should be well approximated by a succession of eigenvalue problems.
Solving the time independent Schrodinger equation for each ,
if , we have the following theorem:
The adiabatic theorem
If varies slowly in time with respect to level spacing, the system prepared in the th eigenstate of will remain in the th instantaneous eigenstate of , picking up only a phase factor.
We say that is the dynamical phase, with the instantaneous eigenenergies.
and is the geometrical phase.
Proof of the adiabatic theorem
Define and via the time independent schrodinger equation,
and treat as a parameter and solve the time independent Schrodinger equation for each .
- For each , forms a complete orthonormal basis.
- Expand solution of full time dependent Schrodinger equation into that basis, for each .
If the Hamiltonian changes with time, then the eigenfunctions and eigenvalues are time-dependant:
But, they are at least orthonormal and complete:
Hence, we can take our most general state, the solution to the time dependant Schrodinger equation
and expand into the instantaneous basis:
Substitute the instantaneous basis into the time dependent Schrodinger equation:
The inner product and orthonormality gives:
Now, take the time derivative of Schrodinger equation
Taking the inner product
Now invoke the adiabatic approximation. Assume is extremely small, and drop the second term. We now have:
the solution is:
And so, the final answer is
Which is the statement of the adiabatic theorem.