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Unformatted text preview: PHY4221: Homework 4 Due Oct. 2, 2008 1. Let A and B be hermitian operators deﬁned by
2 2 A=g 2 ajk |j k| 2 , B=g j =1 k=1 bjk |j k| j =1 k=1 where a11 = −7, a12 = 24, a22 = 7, b12 = i, b11 = b22 = 0.
(a) Calculate the expectation value of A and B if the state is |ψ = |1 .
(b) Calculate the uncertainty product: (∆A)2 (∆B )2 for the same state given in (a)
(c) Calculate the commutator [A, B ], then verify that your result in (b) is consistent with
the uncertainty principle.
2. The operators A and B are deﬁned in Problem (1).
(a) Find the eigenvalues and eigenvectors of A.
(b) Consider that the system is prepared in an initial state |ψ = |1 . If one performs a
measurement of A, what is the probability of getting the outcome corresponding the largest
eigenvalue of A?
(c) To continue part (b) above, if the outcome is indeed the largest eigenvalue of A, then we
make a measurement of B . What is the probability of getting the outcome corresponding
the largest eigenvalue of B ?
(d) To continue part (b) (NOT part (c)), and no matter what the outcome of A is, we make
a measurement of B . What is the expectation value of B ?
(3) If A in problem (1) is the Hamiltonian operator of the system, ﬁnd the state vector at
time t if the initial state is given by: |ψ = |1 . How long does it take the system can return
to its initial state? 1 ...
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