Perturbation theory (PT) comprises an important set of approximation methods in
Quantum Physics. It is traditionally divided into:
The Anharmonic OscillatorWe don't have very much time for these topics, so I have gone through just one example of the first order effect, namely an anharmonic oscillator, using the rare gas Ar as the example. The interaction potential between two Argon (or other closed shell) atoms is often represented by the Lennard-Jones (6-12) potential. This potential has just two parameters, characterizing the well depth and the radius of the minimum, R_{m} or a. In the neighborhood of the minimum, the potential is harmonic, so that the zero order wave functions describing the oscillations of two Ar atoms, mass M, are just those of the SHO, and V_{0}= (1/2)M(wx)^{2}.Away from the minimum, the potential is quite strongly anharmonic, with a cubic term when expanded as a power series in (R/R_{m}-1) which we write as (x/a). So the cubic perturbation potential lV_{1} can be written l(x/a)V_{0}. This cubic term is the main term responsible for thermal expansion. Experimental data for Argon can be found in the handout from chapters 1 and 6 of Rare Gas Solids, vol I. In first order, we can work out the effect of the cubic term and convince ourselves that it is zero because it is antisymmetric (in x). There will be an effect in second order, due to virtual transitions to odd parity excited states. This effect means that the energy of the ground state of the cubic anharmonic oscillator is lower than that of the harmonic oscillator with the same value of V_{0}. If we have a perturbation which is quartic, i.e. lV_{1} = l(x/a)^{2}V_{0}, then there is an effect in first order, with the sign of the energy shift given by the sign of l. |
Relation to Matrix methods: a rotational exampleOn problem set #3, we considered the Hamiltonians of rotating molecules, and the solution of these problems by both Operator and Matrix methods. We recalled that the diagonal matrix elements relate to rotation about the axes x, y and z, and that the off-diagonal elements depend on the differences between such terms. As such, the off-diagonal elements are just those terms which appear in perturbation theory. This topic was discussed in the lectures, with examples from angular momentum quantum numbers 1 (3x3) and 2 (5x5) matrices. By rearranging rows and columns, these matrices can be rearranged into block-diagonal form. From this, one can show the correspondence both with perturbation theory, and with singlets, doublets and triplets, etc., i.e. with spectroscopy in general.This set of issues was the subject of a workshop in class time, and the rotating molecule problem also appears on problem set #5. The specific case of l = 1 and 2 was done in 2001 as a project by Jason Harris, which can be inspected on his website. His description includes a few-line MATLAB5.3 code, which can be downloaded (script1.m) and used to solve rotational problems. The code produces the analytic form of the matrix H, from which you can find the eigenvalues by typing the command eig(H): all very satisfying and simple, especially if you understand what you are doing. |