Dec 3
Jim
I would like you to produce two things by next Monday at the latest:
- finish this (whatever you decide "this" exactly is, something about diagonalizing a perturbed potential in terms of some basis) ... you've been working from Thijssen's book; I suggested a specific problem of a small exp(-x^2) change to a harmonic oscillator, and using the HO basis functions - either one, just be specific, and finish it and write it up
- write an "end of tutorial reflection" on what you studied, what you turned in, what topics were covered, what wasn't covered, and all that. Feel free to make an argument for a specific grade (if this tutorial is for a letter grade.)
- Put links to both those here somewhere.
Alex
This week I progressed into the Hartree-Fock chapter of the Computational Physics by J.M. Thijssen. The approach that Thijssen has taken is to use an arbitrary set of basis functions and linearize a potential in that basis and this is approach he takes with the Hartree-fock method. He first creates a basis and then performs a computation. In some cases, I have a problem expanding on some of the algorithms, like with the linearizing a perturbation in the Quantum Harmonic Oscillator. Using the Griffiths' Introduction to Quantum Mechanics textbook, I found the equations for finding the first perturbation energy given a set of orthonormal basis.
QHO and a gaussian perturbation
Using the gaussian perturbation:
\[\frac{1}{\alpha}e^{\frac{-x^2}{2}}\]
and the equation of the first energy correction to a perturbed potential
$$ E_n^1 = \left< \psi_n \right| H' \left| \psi_n \right>$$
Turning this into a discrete and computable problem then the wave function \( \psi \) becomes a vector of values corresponding to the wave function evaluated at point x where x lies in the space of \( -a \le x \le a \) :
$$ \vec{\psi}_n = \left[ \psi(-a)_n, \psi(a+\Delta h)_n, .... , \psi(a)_n \right] $$
and the perturbation becomes a matrix where diagonal terms are the values of the perturbation at x and all off diagonal terms are zero.
Then the equation for the first energy order correction is:
$$ E^1_n = \vec{\psi}_n \mathbf{H'} \vec{\psi}_n $$
Question about QHO Pert
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