Consequences

Single electron levels

The normalized eigenstates of (9) are plane waves

 

where and ; the corresponding eigenvalues are

 

in analogy with (7) identifying .

Pauli exclusion principle

Since we are now dealing with a gas of identical indistinguishable fermions each single electron level can be occupied by at most one electron. Taking into account the spin of the electron which has two possible values this results in at most two electrons per allowed -vector. So in the ground state (zero temperature) all levels belonging to allowed -vectors contained in some sphere with radius (Fermi sphere), i. e. with energy up to (Fermi energy) are occupied.

Ground-state energy

In contrast to the classical system the quantum system shows a non-vanishing ground-state energy per electron. The velocity corresponding to is typically 10 times larger than the thermal velocity even at room temperature.

Fermi-Dirac-Distribution

As a consequence of the pauli principle the statistics at finite temperature result in a distribution

 

which at typical temperatures drastically differs from the Maxwell-Boltzmann distribution (see figure 1); here , and is the chemical potential which is determined by the normalization condition on the distribution but at room temperature and below equals .

  
Figure 1: Maxwell-Bose vs. Fermi-Dirac distribution; physical constants equal 1, .

Specific heat

For room temperature and below the Sommerfeld theory leads to a specific heat

 

which at room temperature is about 100 times smaller than the Drude result.

DC electrical conductivity, Hall effect, thermal conductivity

All these properties retain their form if one replaces v by since the form of the distribution did not enter the calculations in the Drude theory.

Wiedemann-Franz law

Having performed the steps above the Wiedemann-Franz law takes on the form

 

which is of the same order of magnitude as the Drude result and is in excellent agreement with experimental data at low and high temperature.