Consider a cubical box with rigid walls (i.e., outside of the cube) and edges of length . The general solution for this problem is
,
where , , and are all positive integers. Note that this solution is just the product of three solutions to the onedimensional particle in a box. The energy corresponding to the threedimensional solution is just the sum of the energies for each of the three onedimensional solutions: .
Part A  

What is the smallest allowed energy for a particle in a cubical box? Express your answer in terms of , , and .

Part B  

If you are dealing with a very large , you can assume that each state (point with integer coordinates) corresponds roughly to one unit of volume inside of the sphere. So, the number of states is approximately equal to the volume of the octant of the sphere. Use this idea to find the number of states with energy less than or equal to for a large . 
Express your answer in terms of .
ANSWER: 


Part C  

Suppose that you have particles, each of which has to go into its own state, in the cubical box. Assume that no state is occupied if a lower energy state remains unoccupied. What is the highest energy that one of these particles has? Assume that is large so that the analysis from the previous part applies. 
Express your answer in terms of , , , and .
ANSWER: 


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