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Unformatted text preview: Chem 3890 Physical Chemistry I Fall 2010 Chemistry 3890 Problem Set 6 Fall 2010 Due: in class, Friday, October 22 Last revised: October 16, 2010 Additions/corrections: 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, integraldisplay d x * i i prime = i,i prime (1a) integraldisplay d x * j j prime = j,j prime . (1b) 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 (2) 1. Using only the orthonormality of the states j , and the relations (2), 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 c ij in the expansion j = 3 summationdisplay i =1 i c ij , j = 1 , 2 , 3 (3) To determine these coefficients, note that you can use the orthonormality of the j to determine the integrals ( j | i ) integraltext d x * j i , and that ( | ) = integraldisplay d x * ( x ) ( x ) = bracketleftbiggintegraldisplay d x * ( x ) ( x ) bracketrightbigg * = ( | ) * . (4) 3. If a particle is initially in the state = 1 3 [ 1 + i 2- 3 ] , (5) what is the probability that a measurement of the dynamical variable represented by...
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