ECE 306 Homework Set #1Spring 2005 (Due 12 noon, Wednesday, February 2, 2005.) 1. Consider the following hypothetic wave function for a particle confined in the region 64≤≤−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 >, < x2 > and σ2= < (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 tixeAetxωλψ−−=),(where A, λand ωare positive real constants. a.Normalize Ψb.Determine the expectation values of x and x2c.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 σrepresents the “spread” in x. What is the probability that the particle would be found outside this range?
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