The perturbation theory is best applied in the determination of the approximate correction to the energy levels and eigenstates after a certain perturbation is introduced to a real quantum system. To understand this deeply, let us look at this example.
a) Show that there is no first-order change in the energy levels and calculate the second-order correction.
The first-order change in the energy levels with this given perturbation, H’ = -qEx , is found using the fundamental result of the first-order perturbation theory which states that the change in energy is just the average value of the perturbation Hamiltonian in the unperturbed states:
Substituting the given perturbation into the equation, we get
where n is the nth eigenfunction. Employing the ladder operators (raising and lowering operators, a+ & a–, respectively) on x as in the equation,
and we get the inner product
For the second-order correction, it is found using the fundamental equation of the second order perturbation theory which is
and the above equation becomes
Then, our second-order equation becomes
It is important to note that
b) The Schrödinger equation (SE) can be solved exactly in this case by a change of variables. Find the exact energies and show that they are consistent with the perturbation theory approximation.
The Schrödinger equation for this potential is:
By change of variables, we let
Thus, substituting this to our SE, it becomes,
and rearranging terms, we get
which is the SE for simple harmonic oscillator in the variable x’.
We know that,
In the above equation, the second term is the second order correction to the energy level and since we found that the first order correction is zero, thus this solution is consistent with the perturbation theory approximation.
About the author:
Ann finished her BS Physics degree at MSU main campus in Marawi City and is pursuing now a graduate degree at MSU-IIT, Iligan City. She is into performing experiments in Material Science and hopes to become one of the experimental physicists of the country someday.