Daily_010.SOLN

# Daily_010.SOLN - 2 = 1 2 1 2 cos φ = 1 2 1 2 cos π 2 θ =...

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Phys 274 – c ± Eric B. Szarmes 9 February 2011 Daily Homework 10 SOLUTIONS Q5S.2 We are asked to consider the following series of Stern-Gerlach devices (the y and z directions are drawn perpendicular to the beam direction at the indicated points): SG(y) SG( ! ) + + z y ? z y z y z y " z y + # The quantons enter the second device in the - y spin-state (i.e. with spins pointing in the - y direction), but are forced by the second device to exit either with their spin aligned in the + θ direction (i.e. in the positive channel) or aligned in the - θ direction (i.e. in the negative channel). The choice is determined randomly according to probability. The general rule for the probability to exit the two channels, which applies in all cases , is to look at the angle φ subtended by the input spin and the positive channel of the output device (indicated in the diagram above): the rule then says that the probability to exit the positive channel is equal to P (+ θ ) = cos 2 ( φ 2 ) . (1) We see that φ = π/ 2 + θ . We also have the relation cos 2 ( φ/ 2) = 1 2 + 1 2 cos φ . Therefore, the probability to exit the positive channel is P (+ θ ) = cos 2 ( φ
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Unformatted text preview: 2 ) = 1 2 + 1 2 cos φ = 1 2 + 1 2 cos( π 2 + θ ) = 1 2-1 2 sin θ . (2) For the negative channel, the probability is P (-θ ) = sin 2 ( φ 2 ) = 1 2-1 2 cos φ = 1 2-1 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 = ± a-2 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 e-iθ ² we have h ψ | ψ i = | a e iθ | 2 + | a e-iθ | 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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