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# exam208sol - Name Phys 225/315 Feb 29 2008 75 points...

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Name Phys. 225/315 Feb. 29, 2008 MidTerm Exam 2 BOTH SIDES 75 points Numerical answers may be left in terms of π , 2 , μ N , etc. Show all details of your calculations. Sketches and plots are very helpful. 1. ( 15 pts ) A two-state system has energy levels E a and E b with E a < E b . The system is prepared at t = 0 in a linear combination of the states | E a and | E b so that there is a 25% probability that the energy is found to be E b . a) ( 3 pts ) Write the state of the system | ψ ( t ) at time t . | ψ ( t ) = 3 2 | E a e - iE a t/ ¯ h + 1 2 | E b e - iE b t/ ¯ h b) ( 3 pts ) Is there a time, t , when the system is certain to be found in state | E a ? Always Never Never: since e - iE b t/ ¯ h is never zero. c) ( 3 pts ) What is the first time t (for t > 0) that there is a 25% probability to find energy E b ? Always Never Always: since the time dependence drops out in amplitude squared. d) ( 6 pts ) Derive the expectation value of the Energy E , at time t . Use E a | E a = 1 and E a | E b = 0 Find probability amplitude for each state < E a | ψ ( t ) . When you square this the time dependence drops out. E = 3 4 E a + 1 4 E b This is worked out in detail in McIntyre pages 51 and 52 equations 3.10 to 3.16

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2. ( 15 pts )
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