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Unformatted text preview: Chem 3890 Physical Chemistry I Fall 2009 Chemistry 3890 Problem Set 5 Fall 2009 Due: in class, Friday, October 16 Last revised: October 9, 2009 Problem 1 Consider two quantum mechanical operators A and B , with A | i ) = i | i ) and B | j ) = j | j ) . Take A and B to be Hermitian, so that the respective eigenstates are orthonormal, ( i | i prime ) = i,i prime , ( j | j prime ) = j,j prime . Also, assume that neither operator has degenerate eigenvalues, so that i negationslash = i prime for i negationslash = i prime , and j negationslash = j prime for j negationslash = j prime . Normalized functions | i ) and | j ) are related by the expressions: | 1 ) = 1 2 | 1 ) + 1 2 | 2 ) - 1 2 | 3 ) | 2 ) =- 1 2 | 1 ) + 1 2 | 2 ) | 3 ) = 1 2 | 1 ) + 1 2 | 2 ) + 1 2 | 3 ) (1) 1. Using only the orthonormality of the states | j ) , and the relations (1), verify that the states | i ) are in fact orthonormal. 2. Find expressions for the | j ) in terms of the | i ) . That is, determine the coefficients in the expansion | j ) = 3 summationdisplay i =1 | i )( i | j ) , j = 1 , 2 , 3 (2) To determine these coefficients, first show that, for any 2 states and , ( | ) integraldisplay d x * ( x ) ( x ) = ( | ) * . (3) Hence use the orthonormality of the j to determine expansion coefficients ( j | i ) , and relation (3) to find the ( i | j ) ....
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