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Reif, Problems 7.1, 7.3, 7.5, 7.7, 7.14,^* 7.20

Problem 7:^*
The energy levels of a particle trapped in a cubical box of volume V = L³ are given by
E_n = (²²/2mL²)(n_x ²+n_y²+n_z²). We can use this to model a single atom of helium trapped in a metal crystal. Assume the "box" has sides of length L = 1 angstrom (10^-8 cm).

1. Write down an expression for the partition function, do not attempt to evaluate the series.
2. For T = 300 K, evaluate the first few terms of the partition function
3. Approximate the probability of finding the atom in its first excited state.
4. How big would the box have to be to have a probability near 0.1 of finding the atom in the first excited state?

Note: ^* indicates problems for 517 and 416 honors students only