HW01 & Solutions - ECE 306 Homework Set#1 Spring...

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ECE 306 Homework Set #1 Spring 2005 (Due 12 noon, Wednesday, February 2, 2005.) 1. Consider the following hypothetic wave function for a particle confined in the region 6 4 x : a. Sketch the wave function b. Normalize this wave function over the range the particle is confined in (i.e. find A) c. Determine the expectation values < x >, < x 2 > and σ 2 = < (x - < x >) 2 > using the normalized wave function. d. Again, using the normalized wave function, calculate the expectation value of the kinetic energy of the particle. 2. Consider the wave function t i x e Ae t x ω λ ψ = ) , ( where A, λ and ω are positive real constants. a. Normalize Ψ b. Determine the expectation values of x and x 2 c. Find the standard deviation of x. Sketch the graph of | Ψ | 2 , as a function of, and mark the points (< x > + σ ) and (< x > - σ ) to illustrate the sense in which σ represents the “spread” in x. What is the probability that the particle would be found outside this range?
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