Unformatted text preview: 2 ) = 1 2 + 1 2 cos φ = 1 2 + 1 2 cos( π 2 + θ ) = 1 21 2 sin θ . (2) For the negative channel, the probability is P (θ ) = sin 2 ( φ 2 ) = 1 21 2 cos φ = 1 21 2 cos( π 2 + θ ) = 1 2 + 1 2 sin θ . (3) Q5B.10 To normalize a complex vector  ψ i , the inner product of the vector with itself must equal unity. Thus, for general  ψ i we require h ψ  ψ i = ± ψ 1 ψ 2 ² * · ± ψ 1 ψ 2 ² = ( ψ 1 ) * ( ψ 1 ) + ( ψ 2 ) * ( ψ 2 ) =  ψ 1  2 +  ψ 2  2 = 1 . (4) a) For  ψ i = ± a2 ia ² we have h ψ  ψ i =  a  2 +  2 ia  2 = a 2 + 4 a 2 = 5 a 2 = 1, requiring a = 1 √ 5 b) For  ψ i = ± a (1 + i ) ai ² we have h ψ  ψ i =  a (1 + i )  2 +  ai  2 = ( a 2 + a 2 ) + a 2 = 3 a 2 = 1, requiring a = 1 √ 3 c) For  ψ i = ± a e iθ a eiθ ² we have h ψ  ψ i =  a e iθ  2 +  a eiθ  2 = a 2 + a 2 = 2 a 2 = 1, requiring a = 1 √ 2 In the above normalization procedure, we take the positive square root by convention. 1...
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 Spring '11
 EricSzarmes
 Physics, Work, Plus and minus signs, Database normalization, Complex number, output device

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