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quiz6soln - 1 Problem 1 Do the one-dimensional kinetic...

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1 Problem 1 Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent? ˆ KE = - ¯ h 2 2 m d 2 dx 2 ˆ P = - i ¯ h d dx 1.1 Solution This question requires calculating the commutator of the operators given. h ˆ KE, ˆ P i = - ¯ h 2 2 m d 2 dx 2 - i ¯ h d dx - - i ¯ h d dx - ¯ h 2 2 m d 2 dx 2 = i ¯ h 3 2 m d 2 dx 2 d dx - d dx d 2 dx 2 = i ¯ h 3 2 m d 3 dx 3 - d 3 dx 3 = 0 The operators commute; all done. 2 Problem 2 Given the following wavefunction, describing some quantum particle, ex- panded in a basis of eigenfunctions of a ”location” operator, Ψ = C left ψ left + C right ψ right + C top ψ top + C bottom ψ bottom NOTE :(the eigenfunction ψ left is associated with an eigenvalue of ”LEFT”, and so on for the other eigenfunctions) 2a. What is the probability of observing a value of ”TOP” (before any ex- plicit measurement is made). If we have a detector that measures the ”LEFT” location, and a signal is detected there after a particle is projected towards the array of detectors: 2a. How would you write the expression for the wavefunction? 2b. What is the probability of observing a value of ”LEFT”? 1
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2c. If a measurement was made on the wavefunction 5 hours later, what would be the probability of measuring a value of ”BOTTOM” (assume no
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