Quantum free electron (QFE) theory

QUANTUM FREE ELECTRON (QFE) THEORY

The failures of classical free electron theory were rectified in quantum theory given by Sommerfeld in the year 1928.

This theory uses quantum concepts and hence it is known as quantum free electron theory.

Sommerfeld used Schrodinger's wave equation and de-Broglie's concept of matter waves to obtain the expression for electron energies.

He approached the problem quantum mechanically using Fermi Dirac statistics instead of classical Maxwell - Boltzmann statistics.

Postulates of Quantum Free Electron Theory

• The potential energy of an electron is uniform or constant within the metal.

• The electrons have wave nature.

• The allowed energy levels of an electron are quantized.

• The electrons move freely within the metal and they are not allowed to leave the metal due to existance of potential barrier at its surfaces.

• The free electrons obey Fermi - Dirac statistics.

Merits of Quantum Free Electron Theory

• This theory treats the electron quantum mechanically rather than classically.

• It explains the electrical conductivity, therman conductivity, specific heat capacity of metals, photoelectric effect and Compton effect, etc.

Demerits of Quantum Free Electron Theory

• Even though it explains most of the physical properties of the metals, it fails to state the difference between conductor, semiconductor and insulator.

• It also fails to explain the positive value of Hall coefficient and some of the transport properties of the metals.

Electrons in Metals - Particle in a three dimensional box

The solution of one-dimensional potential well is extended for a three-dimensional potential box.

In a three-dimensional potential box, the particle (electron) can move in any direction in space. Therefore, instead of one quantum number n, we have to use. three quantum numbers, n_x,n_y and n_z, corresponding to the three coordinate axes namely x, y and z respectively.

 If a, b, c are the lengths of the box as shown in figure 2.8 along x, y and z axes, then

Energy of the particle = E_x + E_y + E_z

If a = b = c as for a cubical box, then

The corresponding normalised wave function of an electron in a cubical box may be written as

From the equations (1) and (2), we understand that several combinations of the three quantum numbers (n_x, n_y and n_z) lead to different energy eigen values and eigen functions.

Example

Suppose a state has quantum numbers, then

n_x = 1, n_y = 1, n_z = 2

Then, n_x² + n_y² + n_z² = 6

Similarly, for a combination n^x = 1, n^y =2, n^z = 1 and for acombination n_x = 2, n_y = 1, n_z = 1

we have, n_x² + n_y² + n_z² = 6

E₁₁₂ = E₁₂₁ = E₂₁₁= 6h² / 8ma² ....(3)

The corresponding wave functions are written as