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Unformatted text preview: QUANTUM MECHANICS SUBJECT EXAM Spring 2008 ID NUMBER: 1. POSTULATES OF QUANTUM MECHANICS: A certain quantum system is governed by the Hamiltonian H = n =0 E n | n )( n | , where the state | n ) is the n th energy eigenstate of the system, with eigenvalue E n . Assume that the system is prepared in the initial state | ( t ) ) = 1 6 [ | 1 ) + 2 | 2 ) + | 3 ) ], where the states | 1 ) , | 2 ) and | 3 ) refer to the 1st 2nd and 3rd energy eigenstates. At time t 1 a measurement is made of the total energy of the system. (a) Taking the initial state into account, what are the possible results from the measurement. (b) What is the probability of obtaining each possible result? (c) For each possible result, what is the normalized state of the system after the measurement? Now consider a system of two spin-1/2 particles prepared in the state | ) = a | ) + b | ) + c | ) + d | ) , where | a | 2 + | b | 2 + | c | 2 + | d | 2 = 1. While the particles are in this state, the operator S 2 is measured, where vector S = vector S 1 + vector S 2 is the total spin. (d) What are the possible results of the measurement? (e) What are the probabilities for getting each possible result? (In terms of a , b , c , and d ) ID NUMBER: 2. SCATTERING IN 1-D: A particle with energy E is incident from the left onto a com-...
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