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581hw2sc - AMATH 581 Autumn Quarter 2011 Homework 2 Quantum...

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AMATH 581 Autumn Quarter 2011 Homework 2: Quantum Harmonic Oscillator DUE: 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 five normalized eigenfunctions ( φ n ) and eigenvalues ( ε n ) using a direct method. Be sure to use a forward- and backward-differencing for the boundary conditions (HINT: 3 + 2Δ x KL 2 - ε n 3). For this calculation, use x [ - L, L ] with L = 4 and choose xspan = - L : 0 . 1 : L . Save the absolute value of the eigenfunctions in a 5-column matrix (column 1 is φ 1 , column 2 is φ 2 etc.) and the eigenvalues in a 1x5 vector. NOTE: This procedure solves for the interior points. So be sure at the end to include your first and last 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 that d 2 φ n dx 2 - γ | φ n | 2 + Kx 2 - ε n φ n = 0 . (1) Depending upon the sign of γ , the probability density is focused or defocused. Find the first two
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