Consider 3D cube with N particles. At ground state, calculate the total energy of the system. . Calculate the energy of the ground state of the system and the maximum particle energy called the Fermi energy. Given that there are
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Expert's answer
2019-07-24T15:50:44-0400
Let us consider a gas of N non-interacting fermions in 3D cube of volume V=L3. We will use occupied numbers to describe this system. Set rn=nx2+ny2+nz2 is a radius of "filled states" sphere in n-space.
For the ground state we have nx=ny=nz=1. So, the energy of the ground state is
EGS=2mL2π2ℏ2rn2=2mL23π2ℏ2
The number of states within the radius is
N=2⋅81⋅34πrn3
where we have added a factor of 2 because fermions have two spin states, the factor of 1/8 indicates that we are just using one eighth of the sphere in n-space because all the quantum numbers must be positive.
Then we can relate the Fermi energy to the number of particles in the cube:
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