Question

Discuss Quantization of energy on the basis of wave nature of particle.

Answer 1

Energy quantization

In a wave motion particles come to rest after a distance on half wavelength \((\frac{\lambda}{2}).\) The two ends of this distance must represent a rest position.

When, motion of a particle of mass m is confined to a line of length L, the ends of this length must represent rest position

Hence, L = n.\(\frac{\lambda}{2}.\) here n = 1, 2, 3, ..........

or λ = \(\frac{2L}{n}\)

Thus wavelength is quantized.

The parameter n which takes the integer values 1,2, 3, ......... labels the stationary states in ascending (increasing) order of energy (i.e. as n increases, E increases).

Hence, the energy of free particle of mass m, having motion confined to a line of length L can be only of the following values:

E₁ = \(\frac{h^2}{8mL^2{'}}\)

E₂ = \(\frac{4h^2}{8mL^2{'}}\)

E₃ = \(\frac{9h^2}{8mL^2{'}}\) ............

E₄ = \(\)\(\frac{n^2h^2}{8mL^2{'}}\)

and not any real number between zero and infinite as expected by classical mechanics.

The stationary state with the least energy E₁ = \(\frac{h^2}{8mL^2{'}}\), is called the ground state of quantum system.

Fig. shows the energy level diagram for first five stationary states of a particle of mass m moving freely along a fine of length L.