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KeyHW2 - Homework 2 Key Chemistry 110A 2009 Ilsoo Kim 1 a 0...

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Homework 2: Key Chemistry 110A — 2009 Ilsoo Kim 1. * 0 ( ) ( ) 0 a n m dx x x ϕ ϕ = m n , where ( ) n x ϕ and ( ) m x ϕ eigenfunctions for the particle in a box. * 0 0 0 0 0 ( ) ( ) 2 sin( )sin( ) 2 1 ( ) ( ) cos( ) cos( ) 2 1 ( ) ( ) sin( ) sin( ) ( ) ( ) 1 (sin( ) 0) (sin( ) 0) ( ) ( ) 0 a n m a a a a dx x x n x m x dx a a a n m x n m x dx a a a a n m x a n m x a n m a n m a a a n m n m a n m n m ϕ ϕ π π π π π π π π π π π π = - + = - - + = - - + = - - - + - - + = 2. n x < , where n means the “eigenfuction” of the particle in a box. * 0 0 2 0 0 0 0 ( ) ( ) 2 sin( ) sin( ) 2 sin ( ) 2 1 2 1 cos( ) 2 1 2 cos( ) n L n n L L L L L x dx x x x n x n x dx x L L L n x dxx L L n x dxx L L n x dxx dxx L L ϕ ϕ π π π π π < = = = = - = -
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2 0 0 0 2 2 0 2 2 2 2 1 1 2 2 sin( ) sin( ) 2 2 2 1 2 (2 sin(2 ) 0) cos( ) 2 2 2 1 (cos(2 ) 1) 2 2 1 (1 1) 2 2 2 L L L L L n x L n x x x dx L n L n L L L L n x n n L n n L L L n L n L L L n L π π π π π π π π π π π π = - + = - - - = - - = - - = * 0 0 0 2 0 2 0 ( ) ( ) 2 sin( ) sin( ) 2 sin( )cos( ) 2 1 2 sin( ) 2 2 cos( ) 2 (cos(2 ) 1) 2 0 n L n n L L L L p d dx x x i dx n x d n x dx iL L dx L n n x n x dx iL L L L n n x dx iL L n L n x iL n L n iL ϕ ϕ π π π π π π π π π π π < = = = = = - = - - = h h h h h h This is a general result for any bound eigenstate in quantum mechanics. But this is not always the case if wave-function is prepared in such a way that it is not eigenstate of the system. See 2. (b)
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2 2 2 * 2 0 2 2 2 * 2 0 2 2 2 2 ( ) ( ) ( ) ( ) n L n n L n n p d dx x x dx n dx x x L n L ϕ ϕ π ϕ ϕ π < = - = = h h h , where the orthonormal condition is used for the last step and below calculation for the first step 2 2 2 ( ) 2 sin( ) 2 cos( ) 2 sin( ) ( ) n n d x dx d d n x dx dx L L n d n x L dx L L n n n x L L L L n x L ϕ π π π π π π π ϕ = = = - = -
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