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Unformatted text preview: Dispersive Quantization of the Linear Schr¨odinger Equation Sudarshan Balakrishnan October 3, 2011 Abstract . The study of linear dispersive partial differential equations with piecewise constant periodic initial data leads to quantized structures at times which are rational multiples of π and fractal profiles at irrational multiples of π . We will present an overview of these results for the Schrodinger equation on the torus. § Introduction The purpose of this paper is to study the behavior of the solution to the Schr¨ odinger wave equation with specific initial conditions. First, we go over the methods to analyze the linearly dispersive wave equation established in [5]. Next, we proceed to carry out the same analysis for the Schr¨ odinger wave equation similar to [4] and write programs in MATLAB to plot its solution. We also present an overview of the Fast Fourier Algorithim(FFT) and plot the error involved in the solution to the Schr¨ odinger wave equation. § 1 Linearly dispersive wave equation The linearly dispersive wave equation is an initialboundary value problem on the in terval 0 ≤ x ≤ 2 π given by ∂u ∂t = ∂ 3 u ∂x 3 (1) with initial condition, u ( t, 0) = f ( x ) , (2) and boundary conditions, u ( t, 0) = u ( t, 2 π ) , (3) ∂u ∂x ( t, 0) = ∂u ∂x ( t, 2 π ) , (4) 1 ∂ 2 u ∂x 2 ( t, 0) = ∂ 2 u ∂x 2 ( t, 2 π ) . (5) First, we take the initial data as a unit step function f ( x ) = σ ( x ) = 0 , 0 < x < π ; 1 , π < x < 2 π . (6) The precise values assigned at its discontinuities are not important, although choosing f ( x ) = 1 2 at x = 0 ,π, 2 π is consistent with Fourier analysis. The boundary conditions allow us to extend the initial data and solution to be 2 π periodic functions in x. In order to construct the solution, let us consider a timedependent Fourier Series u ( t,x ) ∼ ∞ X k =∞ b k ( t ) e ikx . (7) As we substitute (7) in equation (1) we get b k ( t ) will satisfy the elementary linear ordinary differential equation: db k dt = ik 3 b k ( t ) . (8) Then, we get: b k ( t ) = b k (0) e ik 3 t . (9) Resubstituting (9) in (7), we get : u ( t,x ) ∼ ∞ X k =∞ b k (0) e i ( kx k 3 t ) . (10) Observe that the solution (10) is 2 π periodic in both t and x . Moreover, since the kth summand is a function of , periodic waves of frequency k move with wave speed k 2 ie., ω ( k ) = k 2 , thereby justifying the dispersive nature of the system. As we solve for the initial value problem, we note the initial condition for the Fourier series given by: f ( x ) ∼ ∞ X k =∞ c k e ikx . (11) where we have : c k = 1 2 π Z 2 π f ( x ) e ikx dx. (12) Now, when we equate u (0 ,x ) = f ( x ), we see that b k (0) = c k which will be the Fourier coefficients of the initial data. The step function (6), has fourier coefficients : 2 b k (0) = , k even, 6 = 0; i πk , k odd ; 1 2 π , k= 0 ....
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This note was uploaded on 11/19/2011 for the course MATH 101 taught by Professor Wormer during the Spring '08 term at UCSD.
 Spring '08
 wormer
 Equations

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