12b-prob3-10

12b-prob3-10 - 1 Ph 12b Homework Assignment No. 3 Due: 5pm,...

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1 Ph 12b Homework Assignment No. 3 Due: 5pm, Thursday, 28 January 2010 1. “A watched quantum state never moves.” Consider a simple model of an atom with two energy levels — the ground state | g a has energy E g and the excited state | e a has energy E e > E g , where ω = ( E e - E g ) / ¯ h ; the Hamiltonian of this system is ˆ H = E g | g aA g | + E e | e aA e | . An experimentalist is equipped to perform a measurement that projects the state of the atom onto the orthonormal basis | + a = 1 2 ( | g a + | e a ) , |-a = 1 2 ( | g a- | e a ) , and to prepare the atom in the state | + a . a ) Suppose that the state | + a is prepared at time 0 and that the measurement projecting onto {| + a , |-a} is performed at time t . Find the probability P t (+) of the + measurement outcome and the probability P t ( - ) of the - measurement outcome. b ) Suppose that the measurement projecting onto {| + a , |-a} is per- formed twice in succession. The state | + a is prepared at time 0, the ±rst measurement is performed at time t , and the second measurement is performed at time 2 t . Find the probability of a + outcome and the probability of a - outcome in the second measurement. c ) Now suppose that N measurements, equally spaced in time, are performed in succession. The state | + a is prepared at time 0, the ±rst measurement is performed at time t , the second measurement at time 2 t , and so on, with the N th measurement performed at time Nt . Find the probability P (+ N ) that the + outcome occurs in every one of the N measurements. d ) For the same situation as in part ( c ), denote the total elapsed time by T = Nt , so that the time interval between the measurements is t = T/N . Show that P (+ N ) can be expressed as P (+ N ) = 1 - f ( ωT ) /N + O (1 /N 2 ) ,
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2 and fnd the Function f ( ωT ). Thus, taking the limit N → ∞ with ωT fxed, we conclude that iF the atom is observed continuously its state never evolves. 2. A better bomb test . The bomb-testing protocol explained in class uses a beam splitter described by the unitary transFormation ˆ U = 1 2 p 1 - 1 1 1 P .
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12b-prob3-10 - 1 Ph 12b Homework Assignment No. 3 Due: 5pm,...

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