disc108sol - h 2 and- h 2 . The average is h 2 P +- h 2 P-=...

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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 -(1-sin )) = h 2 sin Notice what happens when = 0, = 2 , = . Do these results agree with what you expect? Calculate the expectation value of S x | S x | = S x . S x = cos 2 sin 2 h 2 0 1 1 0 cos 2 sin 2 = h 2 (2 cos 2 sin 2 ) = h 2 sin...
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