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3890ps8_09 - Chem 3890 Physical Chemistry I Fall 2009...

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Chem 3890 Physical Chemistry I Fall 2009 Chemistry 3890 Problem Set 8 Fall 2009 Due: in class, Friday, November 13 Last revised: November 6, 2009 Problem 1 A rotor is prepared in the state ψ ( θ,φ ) = N sin 4 [ θ ] exp[ ] . (1) 1. Determine the value of N such that the state ψ is normalized. Note : do not forget the weight factor sin θ in the integration over θ . 2. Use SphericalPlot3D to plot | ψ | 2 for the normalized state. 3. If the z component of angular momentum, ˆ L z , is measured, what values will be found and with what probabilities? 4. If the square of the angular momentum, ˆ L 2 is measured, what values will be found and with what probabilities? [Consider possible lscript values in the range 0 lscript 20.] When evaluating expansion coefficients ( Y lscript,m | ψ ) , use the result of part 3 above in order to decide which m value(s) to use. 5. Calculate the value of ( ˆ L 2 ) for state ψ . Do this two ways: (a) Direct evaluation of ( ψ | ˆ L 2 | ψ ) , using the expression for ˆ L 2 as a differential operator ˆ L 2 = - planckover2pi1 2
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