disc108sol - ¯ h 2 and-¯ h 2 The average is ¯ h 2 P-¯ h...

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Physics 225/315 January 23, 2008 Practice Problems I 1. A beam of atoms is prepared in the state | ψ ± = cos θ 2 | + ± + sin θ 2 |-± a) Send the beam through a SGx device, What are probabilities for x “up” and x “down” after the SGx device? ² + x | = 1 2 ( ² + | + ²-| ) ²- x | = 1 2 ( ² + | - ²-| ) The amplitude for finding the spin in the + x direction is a + = 1 2 ( ² + | + ²-| ) ˆ cos θ 2 | + ± + sin θ 2 |-± ! a + = 1 2 ˆ cos θ 2 + sin θ 2 ! The Probability P + = | a + | 2 which is P + = 1 2 ˆ cos 2 θ 2 + sin 2 θ 2 + 2 cos θ 2 sin θ 2 ! = 1 2 (1 + sin θ ) Similarly P - = 1 2 (1 - sin θ ) Notice that P + + P - = 1 b) What is the average value of the measured x component of spin? Recall that if you measure the x component of spin the possible values are
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Unformatted text preview: ¯ h 2 and-¯ h 2 . The average is ¯ h 2 P +-¯ h 2 P-= ¯ h 2 1 2 (1 + sin θ-(1-sin θ )) = ¯ h 2 sin θ Notice what happens when θ = 0, θ = π 2 , θ = π . Do these results agree with what you expect? Calculate the expectation value of S x ² ψ | S x | ψ ± = ² S x ± . ² S x ± = ‡ cos θ 2 sin θ 2 · ¯ h 2 0 1 1 0 cos θ 2 sin θ 2 = ¯ h 2 (2 cos θ 2 sin θ 2 ) = ¯ h 2 sin θ...
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This note was uploaded on 03/01/2010 for the course PHYSICS 225 taught by Professor Rothberg during the Spring '10 term at University of Washington.

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