Prelim 1 & Solutions

# Prelim 1 & Solutions - ECE 306(Open book 10:10 to 11:00...

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ECE 306 Prelim #1 Spring 2005 (Open book. 10:10 to 11:00 AM) 1. Consider the wave function Ψ (x, t = 0) defined as follows: Ψ (x, t = 0) = x; 0 <= x <= a Ψ (x, t = 0) = 2a – x; a <= x <= 2a Ψ (x, t = 0) = 0; otherwise (a) Sketch Ψ (x, t = 0) (b) Normalize Ψ (x, t = 0) (c) What is the expectation value of x for this wave function? (d) What is the probability that a particle in this state is found in the range 0 <= x <= a / 2? 2. A beam of particles with mass m travel along the x-axis in a zero potential region V(x) = 0. Suppose all the particles are in the same state at t = 0 with the state function given below: ikx ikx e A Ae t x + = = 3 ) 0 , ( ψ (a) Is Ψ (x, t = 0) an eigen function of H? Justify your answer. (b) What is the expectation value of the total energy H in the state Ψ (x, t = 0)? (c) What is the uncertainty in the total energy in the state Ψ (x, t = 0)? (d) Is Ψ (x, t = 0) an eigen function of the linear momentum operator? Explain.

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Unformatted text preview: (e) The linear momentum is measured in the state Ψ (x, t = 0). Find the values that can be measured and the absolute probability of obtaining each result. 3. Consider an electron whose energy eigen functions satisfy H Ψ n = E n Ψ n . The Ψ n ’s are normalized over the 1-D space. Consider a mixed state of the electron at t = 0 given by: 5 2 5 2 ; 5 2 E E ≠ + = (a) Normalize Ψ over the space. (b) Calculate the expectation value of the total energy in state Ψ in terms of E n . (c) If the measurement of total energy is made in the state Ψ , what results can be obtained and with what absolute probability will each of these results be found? (d) What is ∆ E, the uncertainty of the total energy in the state Ψ in terms of E 2 and E 5 ?...
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