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3890ps6_10

# 3890ps6_10 - Physical Chemistry I Chem 3890 Fall 2010...

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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 ˆ A and ˆ B , with ˆ i = α i ψ i and ˆ j = β j φ j . Take ˆ A and ˆ B to be Hermitian, so that the respective eigenstates are orthonormal, integraldisplay d * i ψ i prime = δ i,i prime (1a) integraldisplay d * j φ j prime = δ j,j prime . (1b) 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 (2) 1. Using only the orthonormality of the states φ j , and the relations (2), 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 c ij in the expansion φ j = 3 summationdisplay i =1 ψ i c ij , j = 1 , 2 , 3 (3) To determine these coefficients, note that you can use the orthonormality of the φ j to determine the integrals ( φ j | ψ i ) ≡ integraltext d * j ψ i , and that ( φ | ψ ) = integraldisplay d * ( x ) ψ ( x ) = bracketleftbiggintegraldisplay d * ( x ) φ ( x ) bracketrightbigg * = ( ψ | φ ) * . (4) 3. If a particle is initially in the state χ = 1 3 [ ψ 1 + 2 - ψ 3 ] , (5) what is the probability that a measurement of the dynamical variable represented by ˆ A will yield the value α

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3890ps6_10 - Physical Chemistry I Chem 3890 Fall 2010...

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