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Unformatted text preview: 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...
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This note was uploaded on 03/23/2012 for the course AMATH 581 taught by Professor Staff during the Fall '08 term at University of Washington.
- Fall '08