Wavefunctions for Traveling Electrons

One can easily see that this solution satisfies the boundary conditions 

Putting this back into the Schrodinger wave equation, 

Recall in the previous treatment where we allowed only standing waves, we found energy eigenvalues given by 

Clearly, emplo)fing periodic boundary conditions produces a different set of energy levels however, it will also produce a different density of states. Since E = h k = (2ti/L) h. 

Thus the voliune associated with each energy state in fc-space is 2tt/L). The minimum energy configuration is a Fermi sphere with radius fcmax- The volume of this sphere in fc-space is (4n/3)fc ax- Th number of electrons that can be accommodated is 

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