Daily_010.SOLN - 2 ) = 1 2 + 1 2 cos = 1 2 + 1 2 cos( 2 + )...

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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 = 1 2 + 1 2 cos( 2 + ) = 1 2-1 2 sin . (2) For the negative channel, the probability is P (- ) = sin 2 ( 2 ) = 1 2-1 2 cos = 1 2-1 2 cos( 2 + ) = 1 2 + 1 2 sin . (3) Q5B.10 To normalize a complex vector | i , the inner product of the vector with itself must equal unity. Thus, for general | i we require h | i = 1 2 * 1 2 = ( 1 ) * ( 1 ) + ( 2 ) * ( 2 ) = | 1 | 2 + | 2 | 2 = 1 . (4) a) For | i = a-2 ia we have h | i = | a | 2 + | -2 ia | 2 = a 2 + 4 a 2 = 5 a 2 = 1, requiring a = 1 5 b) For | i = a (1 + i ) ai we have h | i = | a (1 + i ) | 2 + | ai | 2 = ( a 2 + a 2 ) + a 2 = 3 a 2 = 1, requiring a = 1 3 c) For | i = a e i a e-i we have h | i = | a e i | 2 + | a e-i | 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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