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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 + cosHpê2L
2 `p
`
RI 2 j M ØS sinHpê2L
z basis 2  sinHpê2L 1  cosHy Mathematica for Students 1
1
Printed bpê2L
2 2 cosHp ê 2L  2 sinHpê2L
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 + cosHpê2L
2 `p
`
RI 2 j M ØS  sinHpê2L
z basis sinHpê2L
2 cosHp ê 2L 2
1  cosHpê2L
2 sinHpê2L
2 1  cosHpê2L
2  sinHpê2L 2
1 + cosHpê2L
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 spin1 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.
 Fall '13
 AkimasaMiyake

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