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ECE 306 Homework Set #2Spring 2005 (Due 12 noon, Wednesday, February 9, 2005.) 1. Suppose an electron is confined to a zero-potential region between two impenetrable walls at x = 0 and afor all times. Its initial wave function is given by: axxaxaxax≥≤<<=,00,,0)3sin(2{)(πψa.Calculate the expectation value of the energy <^H> in this state b.Calculate the energy eigen value corresponding to this state. c.What is the time dependence of Ψ(x,t)? d.Calculate the uncertainty in the total energy in this state, ∆^H. Is the answer what you expect? Explain. 2. Prove the following commutation relationships: a.[A + B,C] = [A,C] + [B,C] b.[A, BC] = [A,B]C + B[A,C] 3. Prove the following commutation relations: a.[px, xn] = -iÑnxn-1b.[x, px2] = 2iÑpxc.From b above, what can you say about the possibility of measuring the position and kinetic energy of a particle in an arbitrary state simultaneously with zero uncertainty in each measurement?
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