Prelim 1 & Solutions - (e) The linear momentum is...

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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 (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 n = E n n . The 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 over the space. (b) Calculate the expectation value of the total energy in state in terms of E n . (c) If the measurement of total energy is made in the state , 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 in terms of E 2 and E 5 ?...
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Prelim 1 &amp; Solutions - (e) The linear momentum is...

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