Energy states of an electron gas

Let us consider first the simplest case of an electron moving in one dimension, along the x axis. The Schrodinger equation is  

In the particular case under consideration, it is necessary to impose certain conditions on the wave function since the electron is enclosed in the box and may not pass outside it. The wave function must therefore become zero at the surface of the box, and such functions must therefore have an integral number of nodes in the length L. Hence, 

For the gerteral case of three dimensions, the Schrodinger equation has the form  

In each state corresponding to a particular set of values for the three quantum numbers, a maximum of two electrons may be accommodated according to the Pauli principle. Since there arc three quantum numbers it is possible to have different states with the same energy value, i.tf. the states are degenerate. Thus + 3  

All states of equal energy E will lie on a sphere of radius 

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