Graduate Course: Quantum Physics

John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona 

Perturbation Theory examples

Perturbation theory (PT) comprises an important set of approximation methods in Quantum Physics. It is traditionally divided into:
  • Time independent PT, which is subdivided into
    • Non-degenerate PT and Degenerate PT;
  • Time dependent PT.
Each of these sub-cases are associated with detailed formulae which are more or less difficult to learn, and more difficult to remember... There is also the question of the relationship of PT to matrix methods and which method is exact. Each of the two course books, Gasiorowicz and Liboff discuss PT, with examples. Both start with theory of the non-degenerate case, in G3 chapter 11, p174-197, (G2 chap 16 p266-269) and chapter 13.1-2 (L4 p681-700, L3 p702-708) respectively. These sections deal with both first order and second order corrections, for which you have the summary formulae (handout).

The Anharmonic Oscillator

We 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, Rm 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 V0= (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/Rm-1) which we write as (x/a). So the cubic perturbation potential lV1 can be written l(x/a)V0. 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 V0. If we have a perturbation which is quartic, i.e. lV1 = l(x/a)2V0, then there is an effect in first order, with the sign of the energy shift given by the sign of l.

The Van der Waals energy

Sticking with the two Ar atoms, we can ask why these two atoms attract each other, when they are electrically neutral, and have closed shells. In other words, how does the R-6 energy arise? A simple 'classical' explanation is often given in terms of fluctuating dipoles. A dipole moment, p1 on atom(1), gives an electric field E21 at atom(2) which is proportional to p1/R3. This picture of the Van der Waals energy is that the dipole induced on atom(2) reacts back to give a dipole on atom(1), so that the energy (gain) is -p1.E12. But since there is no initial dipole, just a concerted dance of dipole moments and fields, then the energy gain is -p2/R6. In class we discussed the corresponding quantum description, as outlined below.

The ground state of the rare gases is a filled shell with a 1S0 spectroscopic term. The is the 1s2 configuration in He, or the 3s23p6 configuration in Ar. The first dipole-allowed excited state has a 1P1 spectroscopic term, with the 1s2p configuration in He, or either the 3p54s or 3p53d configuration in Ar.

Now consider two separated atoms and their interaction, with a product wavefunction y = y1.y2. Nuclei and electrons interact in both atoms, with a perturbation Hamiltonian (energy)

This formalism is very general but of course complicated. So we expand the potential in a multipole (dipole, quadrupole, etc) series, such that the perturbation H1 is given by

In the quantum case the picture of the dipole is a superposition of ground and excited states, and the Van der Waals energy consists of these "virtual" excited states, which are mixed into the ground state by dipole transitions, in second order perturbation theory. Thus the product ground state wave function yn = y10.y20, and the excited state wave function yk = y1k.y2k', where the excited states k and k' can be different. The second order energy is given by

The resulting matrix elements have angular components based on the Ylm (does this sound familiar?). The radial integrals for the dipole-dipole terms vary as R-6. Dipole-quadrupole terms vary as R-8, and quadrupole-quadrupole terms as R-10. More details as applied to rare gas interactions can be found in the handout from chapter 2 of Rare Gas Solids, vol I.

Relation to Matrix methods: a rotational example

On 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.

Return to Timetable 2, Module 5 or to course home page.

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Latest version of this document: 8th April 2008, ex 14th April 2006.