Chapter4_29 - ) sin( ) sin( 2 ) ( | | ) ( t B t S t y h −...

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PHY4604 R. D. Field Department of Physics Chapter4_29.doc University of Florida Larmour Precession (continued) For an electron (spin ½) at rest in a uniform magnetic field which points in the z-direction, z B B ˆ 0 = r . We have = > + > >= + + + 2 / 2 / / / 0 0 | | ) ( | t B i t B i t iE t iE be ae e b e a t γ χ h h where a and b are arbitrary complex constants with 1 | | | | 2 2 = + b a . The complex constants a and b determine the initial conditions. = >= ) 2 / sin( ) 2 / cos( ) 0 ( | α b a where I have written ) 2 / cos( = a and ) 2 / sin( = b . Hence, >= + 2 / 2 / 0 0 ) 2 / sin( ) 2 / cos( ) ( | t B i t B i e e t Expectation Values: To see what is going on we compute <S x > , <S y > , and <S z > as follows: () () ) cos( ) sin( 2 ) cos( 2 ) 2 / cos( ) 2 / sin( 2 ) 2 / cos( ) 2 / sin( ) 2 / sin( ) 2 / cos( 2 ) 2 / sin( ) 2 / cos( 0 1 1 0 2 ) 2 / sin( ) 2 / cos( ) ( | | ) ( 0 0 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 0 0 0 0 0 0 0 0 t B t B e e e e e e e e t S t t B i t B i t B i t B i t B i t B i t B i t B i x h h h h = = = >= < + + + + Similarly,
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Unformatted text preview: ) sin( ) sin( 2 ) ( | | ) ( t B t S t y h − >= < and ) cos( 2 ) ( | | ) ( h >= < t S t z and hence ( ) z y t x t z S y S x S S z y x ˆ ) cos( ˆ ) sin( ) sin( ˆ ) cos( ) sin( 2 ˆ ˆ ˆ ω + − = > < + > < + > >=< < h r . We see that > < S r is tilted at a constant angle α with respect to the z-axis, and precesses about the direction of the magnetic field ( i.e. z-axis) with the Larmour frequency , B = . x-axis y-axis z-axis α < S > B...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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