AMATH 581 Autumn Quarter 2011Homework 2: Quantum Harmonic OscillatorDUE: Friday, October 21, 2011 (actually at 3am on 10/22)Continuing on the idea of the harmonic oscillator of Homework 1:(b) Calculate the first fivenormalizedeigenfunctions (φn) and eigenvalues (εn) using a direct method. Be sureto use a forward- and backward-differencing for the boundary conditions (HINT: 3 + 2Δx√KL2-εn≈3).For this calculation, usex∈[-L, L] withL= 4 and choosexspan=-L: 0.1 :L. Save the absolute value ofthe eigenfunctions in a 5-column matrix (column 1 isφ1, column 2 isφ2etc.) and the eigenvalues in a 1x5vector. NOTE: This procedure solves for theinteriorpoints. So be sure at the end to include your first andlast point.ANSWERS: Should be written out as A3.dat (eigenfunctions) and A4.dat (eigenvalues)(c) There has been suggestions that in some cases, nonlinearity plays a role such thatd2φndx2-γ|φn|2+Kx2-εnφn= 0.(1)Depending upon the sign ofγ, the probability density is focused or defocused. Find the first two
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