Chapter6_9 - ) ( ) ( 2 2 2 2 2 n p x n x n h and 2 2 3 2 )...

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PHY3063 R. D. Field Department of Physics Chapter6_9.doc University of Florida The Infinite Square Well (3) Average Value of p x : The average value of p x for the n th state is 0 ) cos( ) sin( 2 ) / cos( ) / sin( 2 ) ( ) ( ) ( ) )( ( 0 0 2 = = = Ψ Ψ = Ψ Ψ = > < π θ n L n n n op x n n x d L i dx L x n L x n L n i dx dx x d x i dx x p x p h h h Average Value of p x 2 : The average value of p x 2 for the n th state is 2 2 2 2 2 2 0 2 2 2 0 2 3 2 2 2 2 2 2 2 2 2 2 ) ( sin 2 ) / ( sin 2 ) ( ) ( ) ( ) )( ( L n n L n d L n dx L x n L n dx dx x d x dx x p x p n L n n n op x n n x h h h h h = = = = = Ψ Ψ = Ψ Ψ = > < The Variance ( p x ) 2 : The dispersion for the n th state is 2 2 2 2 2 2 2 ) ( ) ( L n p p p n x n x n x h = > < > =< Uncertainty Relations: We see that for the n th state = 2 1 12
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Unformatted text preview: ) ( ) ( 2 2 2 2 2 n p x n x n h and 2 2 3 2 ) ( ) ( 2 2 h h &gt; = n p x n x n The smallest ( x)( p x ) occurs in the n = 1 state where 2 2 ) 1357 . 1 ( 2 3 2 ) ( ) ( 2 1 1 h h h &gt; = = x p x . Note that these states satisfy the uncertainty relation, but do not reach the absolute lower bound....
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