Final answer:
The average kinetic energy of a conduction electron in a metal at 0 K is (3/5)E due to the Fermi level's cut-off at absolute zero and the parabolic energy-momentum relationship of conduction electrons.
Step-by-step explanation:
Showing Average Kinetic Energy of a Conduction Electron at 0K
To show that the average kinetic energy of a conduction electron in a metal at zero Kelvin is (3/5)E, we start by considering the filled states in the electron energy distribution at 0 K up to the Fermi energy level. The Fermi energy, EF, represents the energy of the most energetic electrons at absolute zero. According to quantum mechanics, no two electrons can occupy the same quantum state, resulting in a distribution of electrons up to this maximum energy level.
Because the temperature is 0 K, all states below the Fermi level are filled, and the distribution is sharp, with a clear cut-off at the Fermi level. The average kinetic energy EAV for electrons within a metal can be calculated by integrating the energy over all of the occupied states and then dividing by the total number of electrons.
As the energy distribution is up to EF, we use the density of states function multiplied by the energy and integrate this from 0 to EF. Using this approach and the fact that the integral of E up to the Fermi level results in EF2/2, the average kinetic energy becomes:
EAV = (3/5)EF
This comes from the fact that the energy of the electrons follows a parabolic relationship to their momentum (E ∝ p2) within the context of the free-electron model, which applies to conduction electrons in metals.