Midterm 3 Solutions

f sinhq 2l z coshq 2l z and multiplying n by

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Unformatted text preview: os q 2 q 2 Setting b± = 1, we calculate the normalization factors a+ 2 a- 2 + b+ 2 + b- 2 = = cos2 Hqê2L sin2 Hqê2L sin2 Hqê2L cos2 Hqê2L +1= +1= 1 sin2 Hqê2L 1 cos2 Hqê2L So imposing that the eigenvectors be normalized, we have up to global phase freedom +n\ ª sinHq ê 2LB ‰- f cosHqê2L sin Hqê2L -n\ ª cosHq ê 2LB- ‰- f sinHqê2L cosHqê2L + z\ + + z\ + -z\F = ‰- f cosHq ê 2L +z\ + sinHq ê 2L -z\ -z\F = -‰- f sinHq ê 2L +z\ + cosHq ê 2L -z\ and multiplying +n\ by a global phase of ‰Â f and -n\ by one of -‰Â f therefore gives the conventional form +n\ = cosHq ê 2L +z\ + ‰Â f sinHq ê 2L -z\ -n\ = sinHq ê 2L +z\ - ‰Â f cosHq ê 2L -z\ Printed by Mathematica for Students MidtermExam3Solutions.nb ü Statistics Mean = 5.50 Stddev = 2.90 Problem 3 (Problem 3.20 on pp 108-109) ü Part a As shown below 1 + cos q 1 2 1, 1n \ Ø8 j, mz \< bais 2 sin q 1 - cos q The fraction of particles transmitted by SGz that are transmitted by SGn is given by X1, 1n 1, 1z \ 2 = 1 2 I 1 + cos q 1 2 sin q 1 - cos q M 0 0 2 2 1 = A 2 H1 + cos qLE Similarly, the fraction of particles transmitted by SGn that are also transmitted by the second SGz is given by 1 + cos q X1, -1z 1, 1n \ 2 = 1 2 H0 0 1L 2 sin q 1 - cos q 2 1 2 = A 2 H1 - cos qLE Therefore, the fraction of particles that survive the third measurement is given by X1, 1n 1, 1z \ 2 X1, -1z 1, 1n \ 2 1 21 2 1 2 = A 2 H1 + cos qLE A 2 H1 - cos qLE = A 4 I1 - cos2 qME = Printed by Mathematica for Students 1 16 sin4 q 5 6 MidtermExam3Solutions.nb ü How to obtain 1, 1n \ As we learnt in problem 3.19 of the homework, there are actually two methods to calculate 1, 1n \ ` ` ` Method 1: E...
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