HW02 & Solutions - ECE 306 Homework Set#2 Spring...

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ECE 306 Homework Set #2 Spring 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 a for all times. Its initial wave function is given by: a x x a x a x a x < < = , 0 0 , , 0 ) 3 sin( 2 { ) ( π ψ a. 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. [ p x , x n ] = -i Ñ nx n-1 b. [ x, p x 2 ] = 2i Ñ p x c. 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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