# inclass1 - The most convenient basis functions φ n x to...

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Computer project 1 PHZ 5156 Results due Tuesday September 16 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 ) = - 3 2 exp " - ³ x - L 2 ´ 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. 1

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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 ±∞
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Unformatted text preview: . 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...
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inclass1 - The most convenient basis functions φ n x to...

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