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The Variational Principle

  The Variational Principle[16, 17] provides the starting point for almost all methods which aim to find an approximate solution to Schrödinger's equation. It may be stated as follows:


Obviously this is extremely valuable because it means that it is always possible to find an upper bound for the ground state energy. It is also possible to use variational methods to study excited states (see chapter gif), but the real strength of this principle lies in finding ground state energies. If a trial function, tex2html_wrap_inline6049 , is guessed for the ground state then the expectation value of the Hamiltonian is


The true normalized eigenfunctions, tex2html_wrap_inline6051 , of tex2html_wrap_inline5881 form a complete basis, so the trial wavefunction, tex2html_wrap_inline6049 , may be expanded as a linear combination of these eigenfunctions,




As the tex2html_wrap_inline6051 are normalised it follows that tex2html_wrap_inline6049 is normalised. Using the expansion of tex2html_wrap_inline6049 to calculate tex2html_wrap_inline6063 from Eq.(gif) gives


where tex2html_wrap_inline6065 is the eigenvalue corresponding to eigenstate tex2html_wrap_inline6051 . Since tex2html_wrap_inline6069 for all n, it is clear that


Variational calculations rely on making a physically plausible guess at the form of the ground state wavefunction, tex2html_wrap_inline6049 , of the Hamiltonian, tex2html_wrap_inline5881 . This guess will be referred to as the trial/guiding wavefunction throughout this thesis. The ``trial'' part of the name refers to the use of the wavefunction as a guess of the true groundstate wavefunction to be used as the input wavefunction in a Variational quantum Monte Carlo (VMC) calculation. The ``guiding'' part refers to the use of the same wavefunction as an input wavefunction in the Diffusion quantum Monte Carlo (DMC) algorithm as part of the mechanism to introduce importance sampling. This will described in more detail in section gif. The trial/guiding wavefunction depends on a number of variable parameters which can be adjusted to minimise the energy expectation value. If the guessed values of these parameters are good and the chosen functional form builds in enough variational freedom to adequately describe the physics of the system being studied, then very accurate estimates of the ground state energy can be obtained. Variational quantum Monte Carlo (VMC) calculations are direct applications of the above variational principle.

next up previous contents
Next: Monte Carlo Methods Up: Quantum Monte Carlo Methods Previous: Quantum Monte Carlo Methods

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996