1. Consider a hypothetical atom with just two energy states: A doubly-degenerate ground state of energy=0eV, and a four-fold-degenerate excited state with energy=0.500eV. Determine the value of the partition function for this system for the following temperatures: (a) in the limit that T-->0K, (b) 300K, (c)  3x10⁴K, (d) 3x10⁵K, (e) 3x10⁶K,  (f) 3x10⁷K, and (g) in the limit the T-->infinity. This last one for T-->infinity should be an exact number.
    
2. Consider a hypothetical atom with just two energy states: A doubly-degenerate ground state of energy=0eV, and a four-fold-degenerate excited state with energy=0.500eV. Find the expression for the average value of the energy <E> of this system. You can do this by hand, or by using Maple or another algebraic program. If you use Maple, etc., then turn in your printout with this problem but I have to see all the steps. In the end, evaluate this at 4,000K.
    
3. Consider a hypothetical atom with just two energy states: A doubly-degenerate ground state of energy=0eV, and a four-fold-degenerate excited state with energy=0.500eV. Find the expression for the heat capacity "C" of this system. You can do this by hand, or by using Maple or another algebraic program. If you use Maple, etc., then turn in your printout with this problem but I have to see all the steps. In the end, evaluate this at 4,000K.
    
4. Evaluate the following integral by hand (i.e., you can not just look it up in a table of integrals, or have Maple or Mathematica to do it) using the Gamma Function G(z) technique that I showed you in class. Refer to Appendix B.2 in your book for a refresher.
       ∫₀^∞e^{-a*x^2}x¹⁷dx.
   Show all of your steps here. Note that, while I'm assigning only one of these problems, you should be able to use this technique for many such problems.
    
5. Calculate the quantity <v⁷> by hand showing all of your steps.
    
6. problem 6.33.
    
7. problem 6.38. I just did something very much like this in class.

That's all for homework #8.

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