The conduction of free electrons inside a conductor is best described by quantum mechanics. However, a classical model makes predictions close to the results of quantum mechanics. It is called the theory of metallic conduction.
In this theory, Newton's second law of motion is used to determine the acceleration of an electron in the presence of an applied electric field. Then, its velocity is expressed via this acceleration.
An electron moves through the crystal, containing positive ions, colliding with the ions. These collisions knock it off in various directions. If there were no applied electric field, the average velocity of the electron would be zero over the collisions. However, in the presence of the electric field, the electron accelerates opposite to the field between the collisions, and hence the average velocity, called the drift velocity, is in the direction of the field.
The average duration between collisions is defined as the mean free time.
The relationship between the current density and the drift velocity relates the current density to the electric field. This relationship is in the form of Ohm's law, from which the material's electrical conductivity can be obtained.
This theory predicts the correct temperature dependence of the conductivity. As the temperature of the material increases, the ion's vibrations increase, and so do the electrons' collisions with these ions. Hence, the mean free time decreases, implying that the conductivity decreases. The prediction that the material's electrical conductivity decreases with increasing temperature is observed in experiments, although the exact dependence can be found by using quantum mechanics only.
