Quantum Free Electron (GFE) Theory

Success of Classical Free Electron Theory

• It is used to verify Ohm's law.

• It is used to explain electrical and thermal conductivities of metals.

• It is used to derive Wiedemann-Franz law.

• It is used to explain the optical properties of metals.

Failures of Classical Free Electron (CFE) Theory

• Classical theory states that all the free electrons absorb the supplied energy. But, the quantum theory states that only a few electrons absorb the supplied energy.

• The electrical conductivity of semiconductors and insulators cannot be explained by this theory.

• The photo-electric effect, Compton effect and black body radiation cannot be explained on the basis of classical free electron theory.

• According to the classical free electron theory, the ratio K /σT is constant at all temperatures. But, it is found that it is not constant at low temperature.

• According to this theory, the value of specific heat capacity of a metal is 4.5R. But, the experimental value is given by 3R. (Here R is the universal gas constant.)

• The susceptibility of paramagnetic material is inversely proportional to temperature. But, the experimental result shows that paramagnetism of metal is independent of temperature. Moreover, ferro-magnetism cannot be explained by this 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, thermal 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, nx, nx and nz, 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 = Ex + Ey + Ez

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 (nx, ny, and nz) lead to different energy eigen values and eigen functions.

Example

Suppose a state has quantum numbers, then

nx = 1, ny = 1, nz = 2

Then,

nx2 + ny2 + nz2 = 6

Similarly, for a combination nx = 1, ny = 2, nz = 1 and for acombination nx = 2, ny = 1, nz = 1

we have, nx2 + ny2 + nz2 = 6

The corresponding wave functions are written as