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**The energy levels of a quantum particle confined to a cubical box are:**

[tex] E = \frac{\hbar^2 \pi^2}{2mL^2} (n_{x}^2 + n_{y}^2 + n_{z}^2) [/tex]

where nx, ny, nz are postiive integers

Define the dimensionless energy

[tex] \epsilon = E \frac{2mL^2}{\hbar^2 \pi^2} [/tex]

and define S(e) to be the number of states less than or equal to e

[tex] E = \frac{\hbar^2 \pi^2}{2mL^2} (n_{x}^2 + n_{y}^2 + n_{z}^2) [/tex]

where nx, ny, nz are postiive integers

Define the dimensionless energy

[tex] \epsilon = E \frac{2mL^2}{\hbar^2 \pi^2} [/tex]

and define S(e) to be the number of states less than or equal to e

**a)Compute S(e) (im going to call it S(e) for all integer values of e from 3 to 300**

suppose e was 3, then S(3) is 1. because if epsilon is 3, then all the n's are 1, and that is ground state

suppose e was 4, then S(4) is 4

e=9, S(5) = 7

e=27, S(6) = 8

do i have to calculate them manually or is there an easier way??

**b)Plot S(e) for all integer values of e from 3, to 300**

trying to figire out the function right now...

**c) One can fit this with a function which is the volume of the positive octant. Derive this expression. It is given by:**

[tex] S (e) = \frac{\pi}{6} e^{3/2} [/tex]

[tex] S (e) = \frac{\pi}{6} e^{3/2} [/tex]

what is a positive octant?? Not sure where to start here...

*Note* I have used epsilon and e intechageably... i am sorry if it causes any confusion.*

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