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ps 3 604

# ps 3 604 - ECE 604 LINEAR SYSTEMS Problem Set#3 Issued...

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ECE 604 LINEAR SYSTEMS Problem Set #3 Issued: Thursday, October 1, 2009 Due: Thursday, October 8, 2009 Problem 1: a) Consider the homogeneous system: ! x t ( ) = 0 e t e ! t ! 1 " # \$ \$ % & x t ( ) x 0 ( ) = 1 1 " # \$ % & (1) Define P t ( ) = e ! t 1 ! 1 e t " # \$ \$ % & Let z t ( ) = P t ( ) x t ( ) . Find a differential equation (including initial conditions) for z t ( ) for t ! 0 . b) Compute the transition matrix for (1). Problem 2: One of the simplest problems in quantum mechanics is the one dimensional potential well problem . The well potential is shown in Figure 1. Here, the wave function ! satisfies d 2 dx 2 ! x ( ) " # 2 ! x ( ) = 0 x > a d 2 dx 2 ! x ( ) + " 2 ! x ( ) = 0 x # a Assume that ! and ! are real, and that ! and d ! dx are continuous at x = ± a . Determine conditions on ! and ! such that there exists a solution ! which goes to zero at ± ! . Sketch ! x ( ) as a function of x . Note: ! is the response of a homogeneous linear system in each of the three regions. Match the initial conditions of the three responses using the continuity assumptions. (Although the system for x ! " a is non-causal, you can replace t by ! " to get a system that is causal in ! . After finding the solution, you can back-

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ps 3 604 - ECE 604 LINEAR SYSTEMS Problem Set#3 Issued...

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