3890ps5_09 - Physical Chemistry I Chem 3890 Fall 2009...

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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 ˆ A and ˆ B , with ˆ A | ψ i ) = α i | ψ i ) and ˆ B | φ j ) = β j | φ j ) . Take ˆ A and ˆ B to be Hermitian, so that the respective eigenstates are orthonormal, ( ψ i | ψ i prime ) = δ i,i prime , ( φ j | φ j prime ) = δ j,j prime . Also, assume that neither operator has degenerate eigenvalues, so that α i negationslash = α i prime for i negationslash = i prime , and β j negationslash = β j prime for j negationslash = j prime . Normalized functions | ψ i ) and | φ j ) are related by the expressions: | ψ 1 ) = 1 2 | φ 1 ) + 1 2 | φ 2 ) - 1 2 | φ 3 ) | ψ 2 ) = - 1 2 | φ 1 ) + 1 2 | φ 2 ) | ψ 3 ) = 1 2 | φ 1 ) + 1 2 | φ 2 ) + 1 2 | φ 3 ) (1) 1. Using only the orthonormality of the states | φ j ) , and the relations (1), verify that the states | ψ i ) are in fact orthonormal. 2. Find expressions for the | φ j ) in terms of the | ψ i ) . That is, determine the coefficients in the expansion | φ j ) = 3 summationdisplay i =1 | ψ i ) ( ψ i | φ j ) , j = 1 , 2 , 3 (2) To determine these coefficients, first show that, for any 2 states φ and ψ , ( ψ | φ ) ≡ integraldisplay d * ( x ) φ ( x ) = ( φ | ψ ) * . (3) Hence use the orthonormality of the φ j to determine expansion coefficients ( φ j | ψ i ) , and relation (3) to find the ( ψ i | φ j ) . 3. If a particle is initially in the state | χ ) = 1 3 [ | ψ 1 ) + i | ψ 2 ) - | ψ 3 ) ] , (4) what is the probability that a measurement of the dynamical variable represented by ˆ A will yield the value α 2
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