Degenerate States and Tunneling

DEGENERATE STATES

It is seen from equations (3) and (4) that for several combinations of quantum numbers, we have the same energy eigen value but different eigen functions. Such a state of energy levels is called degenerate state.

The three combinations of quantum numbers, (112), (121) and (211), which give the same eigen value but different eigen functions are called 3-fold degenerate state.

Non-degenerate state:

When only one wave function corresponds to the energy eigen value, such a state is called non-degenerate state.

Suppose n_x = 2, n_y = 2, n_z = 2

TUNNELING

i. According to classical ideas, a particle striking a hard wall has no chance of leaking through it. But, the behaviour of a quantum particle is different due to the wave nature associated with it.

ii. We know that when an electromagnetic wave strikes at the interface of two media, it is partly reflected and partly transmitted through the interface and enters the second medium.

iii. In a similar way the de Broglie wave also has a possibility of getting partly reflected from the boundary of the potential well and partly penetrating through the barrier.

iv. Fig. 2.9 shows a particle with energy E< V approaching potential barrier of height V.

v. An electron of total energy E approaches the barrier from the left. From the view-point of classical physics, the electron would be reflected from the barrier because its energy E is less than V.

vi. For the particle to overcome the potential barrier, it must have an energy equal to or greater than V.

Quantum mechanics leads to an entirely new result. It shows that there is a finite chance for the electron to leak to the other side of the barrier.

It is noted that the electron tunneled through the potential barrier and hence in quantum mechanics, this phenomenon is called tunneling.

The transmission of electrons through the barrier is known as barrier penetration.