Unformatted text preview: PHY4221: Homework 6 Due Oct. 16, 2008 1. A conﬁned electron interacts with a uniform magnetic ﬁeld in the xdirection (i.e., B =
B0 x).
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(a) A spin measurement can give a +1 result if the state is θ = cos θ ↑ + i sin θ ↓ , and
−1 if the state is orthogonal to θ . Construct a Hermitian operator A corresponding to the
measurement above.
(b) Let the initial spin state be ↑ at t = 0, and the system evolves in the presence of the
ﬁeld. If A is measured at time T , what is the expectation value? The measurement of A
takes place in a very short time, which is practically zero. Note: It is more convenient to
use Schr¨dinger picture in measurement problems.
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(c) Let the initial spin state be ↑ at t = 0, and A is repeatedly measured at times t = nT
(n = 1, 2, ....) as long as the results of each measurement is +1, but the process stops if the
ﬁrst −1 result appears. Show that if θ = 0 and T is suﬃciently short but nonzero, then
the probability of having the process stop approaches zero. This is called the quantum Zeno
eﬀect, which means that the system remains in its initial state ‘forever’. How short the T is
needed?
2. A Hamiltonian of a particle with two internal states is given by ∆f H=
f0
(a) Find the eigenvalues and eigenvectors of the Hamiltonian. Then if ∆ f , show that the eigenvectors of the system is almost the aﬀected by f , but the eigenvalues are shifted
by ±f 2 /∆.
(b) Based on (a), suggest a way to trap a particle in position space.
3. A simpliﬁed model of neutrino oscillations can be understood by a twostate system in
which two ﬂavor states are denoted by: ν1 = cos θ m1 + sin θ m2 and ν2 = cos θ m2 −
sin θ m1 , where m1 and m2 are states with masses m1 and m2 respectively. Show that
ν1 can become ν2 if mj are free (noninteracting) states with m1 = m2 . How does the
oscillation period depend on the masses m1 and m2 in the relativistic limit? 1 ...
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.
 Spring '11
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