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Unformatted text preview: 1 Problem 1 Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent? KE =- h 2 2 m d 2 dx 2 P =- i h d dx 1.1 Solution This question requires calculating the commutator of the operators given. h KE, 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 = 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, = C left left + C right right + C top top + C bottom bottom NOTE :(the eigenfunction 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?...
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