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?