comp3 - . 1 The most convenient basis functions φ n ( x )...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Computer project 3 PHZ 5156 Results due Tuesday October 10 Please submit your code and plots wherever requested. Results can be handed in either as a hardcopy, or as an electronic document (e.g. tex, latex, MS word, or even a .pdf) sent via email. Consider the energy eigenstate problem, ± - ~ 2 2 m d 2 dx 2 + V ( x ) ² ψ ( x ) = ( x ) with the boundary conditions ψ ( x = 0) = ψ ( x = L ) = 0. Take L = 10 and for the potential energy V ( x ), V ( x ) = - 0 . 1 exp [ - 5( x - 5) 2 ] 1. So that the expansion in a truncated basis ψ ( x ) nmax - 1 X n =1 c n φ n ( x ) (where this becomes an equality in the limit where nmax approaches infinity), the eigenstate problem can now be defined by Hc = Ec where c is a column vector of the expansion coefficients c n and H is a square matrix with elements of H are given by H mn = - ± n δ mn + V mn for n, m = 1 , 2 , 3 , 4 , ... . The elements V mn , which can be found analytically and stored, are given by the expression V mn = Z L 0 φ * m ( x ) V ( x ) φ n ( x ) dx. It also turns out that V ( x ) is practically zero at x = 0 and x = L , so when you perform an analytic computation of the integral, you may replace the limits by ±∞
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . 1 The most convenient basis functions φ n ( x ) to use are the ones that are the eigenstates of the particle in a box problem, φ n ( x ) = r 2 L sin ± nπx L ² In this basis, we have ± n = ~ 2 n 2 π 2 2 mL 2 2. Solve the problem above using a computer program that diagonalizes the matrix H . Give the lowest five eigenvalues and plot the lowest five wave functions ψ ( x ). Experiment for different values of nmax . At what point do the lowest five eigenvalues not depend strongly on nmax ? Use (as usual) ~ = m = 1. 3. Use a numerical implementation of perturbation theory to first and second order to estimate the eigenvalues. Compare to the exact results above. Does this tell you whether the perturbation can be regarded as small? Note that the elements of your matrix H mn already contains the terms that are needed to do the perturbation theory estimate. 2...
View Full Document

This note was uploaded on 08/08/2011 for the course PHZ 5156 taught by Professor Johnson,m during the Fall '08 term at University of Central Florida.

Page1 / 2

comp3 - . 1 The most convenient basis functions φ n ( x )...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online