Graduate Course: Quantum Physics

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

The Variational Principle and some applications

The Variational Principle

The variational principle provides an alternative approximation method to Perturbation theory, which is particularly powerful for finding ground state energies. It is based on one of the points we have already used in PT, namely that an accurate estimate of the energy can be obtained using a less-accurate wavefunction.

From the expansion theorem, we know that the expectation value of the Hamiltonian is the sum of the eigenenergies En, each weighted by the probabilities |cn|2. In deriving the variational principle, we replace all the En by E0. This substitution means that the true value of E0 is bounded by

where, typically, the wave function is a function of one or more parameters. The variational theorem implies that one can introduce more parameters into the wave function, differentiate to find the 'best' wave function of that particular form, and thereby get closer and closer to the true ground state energy.
 

Examples using the Variational Principle

There are very many examples of the use of the variational principle, several of which are suitable for further reading or for projects. In class we have worked through the classic case of finding the ground state energy of the helium atom. This problem is studied in all textbooks, and in more detail in Robinett (1997) as given in the reference list. The outline of this argument is given below. Additional examples we discussed in class include:
  • The shape of the nuclear potential and the role of spin-orbit coupling in determining "magic numbers" of neutrons and protons in nuclei. This is standard material for a nuclear physics class, and reference was made (handouts) to Williams (1990), chapters 4 and 8.
  • The development and use of Density Functional Theory (DFT) in the Local Density Approximation (LDA). This is a general method for finding the ground state energy of complex systems, notably clusters and solids, which resulted in the 1998 Nobel Prize for Chemistry being awarded to John Pople and Walter Kohn. More about this topic can be found in lecture 4 of my Quantum mechanical models of solids course, given at Sussex University since 1999. This lecture includes a project on jellium, which is also in my book, chapter 6. OK, all publicity is good publicity, and why do something again when you got it right the first time?
 

The Helium atom

The classic example of the application of the variational principle is the Helium atom. We have to take into account both the symmetry of the wave-function involving two electrons, and the electrostatic interaction between the electrons. These two topics were discussed in the lectures, starting from Helium (Z = 2), viewed as a pseudo-Hydrogen atom, when the interaction between the electrons is neglected. Symmetry alone forces the introduction of single and triplet states of Helium, termed para- and ortho-helium respectively. However, with Z=2 and no interaction one finds that the ground state has an energy of -108.8 eV (i.e. -2x4x13.6 eV), whereas the experimental value is -78.975 eV, as given by Gasiorowicz. More accurate experimental values are taken from Heckmann and Träbert (1980) and Robinett (1997), as given in the reference list.

From this starting point we can introduce the electron-electron (EE) interaction as a first order perturbation, and show that the agreement with experiment is improved. However, the EE term (~ +34 eV) is really far too large for us to expect PT to give a good answer. The next stage is to consider an 'effective He atom', with Z* < Z, where the reason Z* is less than Z is that the electron-nucleus interaction is screened by the presence of the other electron.

This description is given in many textbooks, and in outline by Gasirowicz (G3 p224-6, G2 p302-4). He shows that, if Z* is the sole adjustable parameter in the hydrogenic wavefunction, we need to minimize an energy expression containing three terms,

with respect to Z*. This gives simply Z* = Z - 5/16. The correponding energy is now -77.38 eV, much closer to experiment than first order perturbation theory acting on its own. The 5/16 screening value is interesting, as it indicates that the first electron is in effect unscreened, and the second electron is partially screened. Take enough time with the above equation in a book description to be clear how the individual terms arise.

Subsequent developments of the variational principle increased the number of parameters to get better fits to experiment, starting in the case of Helium with a paper by Hylleraas in 1930, with 6 terms, and culminating in 1962 with Pekeris who used 1078 terms, getting closer to the experimental result in the process, not always monotonically. This is a consistent feature of the variational principle in practice; more terms always give lower values, and hence, if all the important components are present in the model, better agreement with experiment.

But one should perhaps be wary of seeking too much meaning in the details of such improvements; a good qualitative understanding can often best be obtained from a simple model, plus the statement that the refined energies can be obtained by allowing the parameters (often associated with the detailed shape of the wavefunction) to be optimized according to the variational principle.


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Latest version of this document: 10th April 2008.