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Perturbation and variation methods

Abderrahmane Al jazairi January 17, 2020

Perturbation theory

  • Best for small changes to a known system.
  • Hamiltonian is modified.
\hat{H}=\hat{H}_0+\lambda\hat{H}_1
The unperturbed Hamiltonian of a known system is modified by adding a perturbation with a variable control parameter λ, which governs the extent to which the system is perturbed.
The perturbation can affect the potential, the kinetic energy part of the Hamiltonian, or both. As an example, consider a double well potential created by superimposing a periodic potential on a parabolic one. This might apply e.g. to a defect in a crystalline lattice.
Fig.: Double well potential.
The Schrödinger equation for the perturbed system is
\hat{H}\psi_n=E_n\psi_n,
that for the unperturbed (known) system is
\hat{H}_0\psi_n^0=E_n^0\psi_n^0.
The index n just serves to identify a particular wave function (e.g. one which uses the same quantum number both for the perturbed and the unperturbed variant).

Variation principle

  • Best for combining systems of comparable weighting.
  • Wave function is modified.
\psi=c_1\psi_1+c_2\psi_2
Two (or more) wave functions are mixed by linear combination. The coefficients c1, c2 determine the weight each of them is given. The optimum coefficients are found by searching for minima in the potential landscape spanned by c1 and c2.
Fig.: Potential landscape.
The energy minima are found by finding the differentials \frac{\partial E}{\partial c_1}=\frac{\partial E}{\partial c_2}=0 and setting them to zero.
Given that
E=\int\psi^*\hat{H}\psi{\rm d}x=(c_1\psi_1^*+c_2\psi_2^*)\hat{H}(c_1\psi_1+c_2\psi_2),
we see that the energy eigenvalue has separate contributions coming from ψ1 or ψ2 only:
=c_1^2\int\psi_1^*\hat{H}\psi_1{\rm d}x+c_2^2\int\psi_2^*\hat{H}\psi_2{\rm d}x\,+\cdots
and a cross term known as the overlap integral:
\cdots+\,c_1c_2(\int\psi_1^*\hat{H}\psi_2{\rm d}x+\int\psi_2^*\hat{H}\psi_1{\rm d}x).
The overlap integral causes the difference in energy between bonding and anti-bonding molecular states.


Reference: http://users.aber.ac.uk/ruw/teach/237/perturb.php

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