Chapter 3 Problem Solutions

Ri 2 j m y 2 using the representation of rhq j l

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Unformatted text preview: 2 3Â = 2 2 7 — ü Part c ` The probability that a measurement of Sx will yield the value — for this state is given by 2 X1, 1x y\ = `† p Z1, 1z R I 2 j M y^ 2 = `p Z1, 1z RI- 2 j M y^ 2 ` Using the representation of RHq j L derived in problem 3.19, we have 1 + cosH-pê2L 2 `p ` RI- 2 j M ØS sinH-pê2L z basis 2 - sinH-pê2L 1 - cosHy Mathematica for Students 1 1 Printed b-pê2L 2 2 cosH-p ê 2L - 2 sinH-pê2L 2 = - 2 1 2 0 1 2 1 2 ` The probability that a measurement of Sx will yield the value — for this state is given by 2 X1, 1x y\ = `† p Z1, 1z R I 2 j M y^ 2 `p Z1, 1z RI- 2 j M y^ = Ch3ProblemSetKey.nb 5 2 ` Using the representation of RHq j L derived in problem 3.19, we have 1 + cosH-pê2L 2 `p ` RI- 2 j M ØS - sinH-pê2L z basis sinH-pê2L 2 cosH-p ê 2L 2 1 - cosH-pê2L 2 sinH-pê2L 2 1 - cosH-pê2L 2 - sinH-pê2L 2 1 + cosH-pê2L 2 1 2 = - 1 1 2 2 1 1 2 1 0 2 - 2 1 2 1 2 Thus 1 2 2 X1, 1x y\ 1 = H1 0 0L - 14 1 2 1 1 2 1 2 1 = 2 J = X1, 1x y\ 1 14 1 14 J2 + = 1 14 1 2 2 1 9 J2 + 1 N2 3Â 2N+ 1 0 2 - 1 2 1 2 2 1 2 1 2 3Â 2 2 3Â 2 2 2N ü Alternative method ` One may also obtain X1, 1x by solving the eigenvalue problem for Sx , as ` Sx 1, 1x ^ = +— 1, 1x ^ — 2 ` of Sx found in (b) 010 101 010 v ¬༼ Let 1, 1x \ ` ØS basis z u v w u v and use the representation w u =— v w =u 2 Hu + wL 2 v 2 =v fl u=w= v 2 =w and imposing normalization gives ` Z1, 1x ØS z 1 basis 2 I1 2 1M ` `p up to a global phase. As we expect, this is the first row of the Sz -basis representation of RI- 2 j M given above, so we are guaranteed the same result for X1, 1x y\ 2 . Printed by Mathematica for Students 6 Ch3ProblemSetKey.nb Problem 3.19 We seek the state 1, 1n \ of a spin-1 particle, for which Sn = —, where n = sin q i + cos q k. We first note that Âq ` `† ` ` ` -S ¬༼ Insert R Hq j L RHq j L = 1ཽ and apply RHq j L = ‰ — y ` Sz 1, 1z ^ = — 1, 1z ^ ` ` `† ` ` RHq j L Sz R Hq j L RHq j L 1, 1z \ = — RHq j L 1, 1z \ ` Sn `` ` Sn BRHq j L 1, 1z \F = — BRHq j L 1, 1z \F so we s...
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This note was uploaded on 02/01/2014 for the course PHYC 491/496 taught by Professor Akimasamiyake during the Fall '13 term at New Mexico.

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