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HW6 - CHM 374 Spring 2010 HOMEWORK SET 6 Due Fri March...

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CHM 374 Spring 2010 HOMEWORK SET 6 Due Fri, March 5th (in class) Read Understanding Physical Chemistry ( UPC ), Chapter 6, Section 6.5 1. Use the matrix representations of the harmonic oscillator Hamiltonian (Eq. 6.33) and eingenvectors (Eq. 6.35) to verify that ) 2 ( ) 2 ( 2 ˆ 2 2 1 2 1 2 h H E . 2. Use the matrix representations of n ˆ , and a ˆ (Eqs. 6.33, 6.36, and 6.37) to confirm that n a a ˆ ˆ ˆ and 1 ˆ , ˆ a a . Note that the number 1 may be equated with a diagonal unit matrix (whose diagonal elements are all equal to 1, and off diagonal elements are all equal to zero). 3. Consider a particle in a cubic 3D box (with L x = L y = L z = L). a. What is the degeneracy of the energy level for which the three quantum numbers (n x , n y , and n z ) are 1, 2, and 3 (combined in any order)? b. Compare the above degeneracy with the fact that the number of permutations of n different (distinguishable) objects is n!.
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