Ground-state energies of electron gas

[tsopa]

I read from a textbook that there are two boundaries conditions that may be used in order to determine the energies of N electron system in a cube of volume V (and side a).
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(check out the attached file)
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As you can see in the attached file, the energies got by using the the two boundary condition methods are different.

The textbook however states that these energies are not different but are EQUAL if proper values of n₁, n₂, n₃ are taken. How can that be possible? I just see two different things that cannot be equal here. Please help.

Notice that n_i = 1, 2 ,3 , ... in the first boundary condition and
n_i = ... -2, -1, -1,0 , 1, 2 ,3 , ... in the second condition.
(i = 1,2,3)

 

[TSny]

The set of energy levels of the individual states are in fact different for the two different boundary conditions. However, if you put N fermions in the box and calculate the total ground state energy of the system, you will get the same total energy for the two different boundary conditions. This is due to the different restrictions on the values of the n_i's for the different boundary conditions.

(There's a factor of pi^2 missing in your expression for the energy levels for the periodic boundary conditions.)

 

[tsopa]

Thanks a lot. I got it.