Q. What is quantum mechanical "Tunneling"? Give one example.

OR

What do you mean by quantum mechanical tunneling? Show that the tunneling probability is given by the expression.

Where U0 =Height of the rectangular potential barrier. Draw graphs showing variation of T with particle energy E and barrier width a.

OR

With help of suitable diagrams explain the phenomenon of quantum mechanical tunneling in a- decay process.

Ans.

Tunnel Effect : Tunnel effect is purely a quantum mechanical phenomenon, absolutely inconceivable in classical physics. The reason is that a particle "in the tunnel" ought to have a negative kinetic energy. In quantum mechanics, the division of the total energy into kinetic and potential energies has no sense because it contradicts the uncertainty principle. If we say that a particle has a definite kinetic energy K it would mean that it has a definite momentum. Similarly, as a particle with a definite location in space. Since the momentum and location of a particle cannot simultaneously have definite values, it is impossible to find simultaneously exact values of kinetic and potential energies. Thus, although the total energy E of a particle has a quite definite value, it cannot be represented in the form of the sum of the exactly determined energies K and V. Hence in quantum mechanics the  conclusion that K is negative inside the tunnel is meaningless.

Application of Tunnel Effect - (Theory of a - decay)

The tunnel effect has been applied to explain the emission of particles from radioactive nuclei. The average energy of an a particle formed within the nucleus is less than the height of the potential barrier around the nucleus which is formed by the nuclear binding forces. Classically, the a-particle cannot escape from the nucleus, but quantum mechanically it "tunnels" through the barrier. This tunneling explains radioactive a decay.

The tunnel effect has also been found responsible for the "field emission" of electrons form metals. This is the emission of electrons by metals in vacuum subjected to strong electric fields. Classically, the field must be much more stronger than the actual fields at which the emission actually occurs. The actual field, however, reduces the width of the barrier for electrons at the metal vacuum interface, so that electrons of energy less than the barrier height can "tunnel" through the barrier.