Intermediate Quantum Mechanics I Midterm exam 1,Fall 2014 n cos Z e i sin Z ,in 2 2 correspondence with the unit vector n sin cos i sin sin j cos k ,which specifies a The state of a spin-1/2 particle may be parametrized as point on a unit sphere in terms of polar angle and azimuthal angle . z y x Problem1 The state is given as 1 i 3 Z Z . 2 2 (1) Determine a particular axis vector n (namely and ),by comparing with n . (2) If one measures the spin-1/2 along the n direction,what will be the outcome(s)? (3) Calculate the expectation value S Z and its uncertainty S Z (4) Calculate the expectation value S X and its uncertainty S X Problem2 A beam of spin-1/2 particles is sent through 3 sequential Stern-Gerlach measurement devices as SZ N 2 Sn SGn SGZ 2 SGZ SZ 2 (1) When the middle axis vector n j (namely the y axis),what fraction of the N particles transmitted through the first device will survive the third measurement? (2) For a general axis n ,what is the state n which represents outcome of the second 2 measurement along the n direction? (3) For a general axis n ,calculate the fraction of the particles which survives the third measurement in terms of and . (4) What is the maximum of the fraction calculated in (3),and when is it achieved? (5) What is the minimum of the fraction calculated in (3),and when does it happen?

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