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Midterm - when suddenly the spring constant quadruples so =...

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QUANTUM MECHANICS PHYSICS 446A Mid-term Exam October 20, 2008 Instructions : Answer all four questions. Closed book exam. Calculators are not permitted. This exam comprises 3 pages. Time allotted is 2 hours. Write down as much information as possible to clearly describe your reasoning. If you do not have time to make some calculations, explain in words how you would go about solving the problem, and hence get partial marks. 1. (8 pts ) A particle in a simple harmonic potential well starts out with a wave function given by the following combination of the ground state 0 and third excited state 3 :  x , 0 = 1 2  0 x  3 x  . (a) What is the total wave function  x ,t as some later time? Express your answer in terms of the symbols 0 , 3 , and . (b) What is the expectation value of energy as function of time t ? 2. ( 8 pts ) A particle is in the ground state of the harmonic oscillator with classical frequency
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Unformatted text preview: , when suddenly the spring constant quadruples, so ' = 2 , without initially changing the wave function (of course, will now evolve differently, because the Hamiltonian has changed). (a) What is the probability that a measurement of the energy would still return the value ℏ / 2 ? (b) What is the probability of getting ℏ ? 3. ( 10 pts ) Use the separation of variables in cartesian coordinates to solve the infinite cubical well (or “particle in a box”): V x, y,z = { if x, y,zare all between anda ∞ otherwise (a) Find the stationary states, and the corresponding energies. (b) Call the distinct energies E 1, E 2, E 3,. .. , in order of increasing total energy. Find E 1, E 2, E 3, E 4, E 5 and determine their degeneracies (that is, the number of different states that share the same energy). 1...
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