qmhw6 - PHY4221 Homework 6 Due Oct 16 2008 1 A confined...

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Unformatted text preview: PHY4221: Homework 6 Due Oct. 16, 2008 1. A confined electron interacts with a uniform magnetic field in the x-direction (i.e., B = B0 x). ˆ (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. o (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 non-zero, 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 two-state 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 (non-interacting) 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.

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