de-Broglie Relation and Significance of de-Broglie waves

de-Broglie Relation

The wavelength of the wave associated with any material particle was calculated by analogy with photon as follows :-

In case of a photon, if it is assumed to have wave character, its energy is given by

E = hv (according to the Planck's quantum theory) ...(i) where v is the frequency of the wave and h is Planck's constant.

If the photon is supposed to have particle character, its energy is given by

E = mc² (according to Einstein equation) ...(ii)

where m is the mass of photon and c is the velocity of light.

From equations (i) and (ii), we get

h ν = mc²

But ν= c / λ

h . c / λ = mc²

or λ = h / mc

de Broglie pointed out that the above equation is applicable to any material particle. The mass of the photon is replaced by the mass of the material particle and the velocity 'c' of the photon is replaced by the velocity v of the material particle. Thus, for any material particle like electron, we may write

λ = h / mv or λ = h / p

where mv = p is the momentum of the particle.

The above equation is called de Broglie equation and 'λ' is called de Broglie wavelength.

Thus the significance of de Broglie equation lies in the fact that it relates the particle character with the wave character of matter.

Louis de-Broglie's concept of dual nature of matter finds application in the construction of electron microscope and in the study of surface structure of solids by electron diffraction. The de-Broglie's concept can be applied not only to electrons but also to other small particles like neutrons, protons, atoms, molecules etc.,

Significance of de-Broglie waves

The wave nature of matter, however, has no significance for objects of ordinary size because wavelength of the wave associated with them is too small to be detected. This can be illustrated by the following examples.

i)          Suppose we consider an electron of mass 9.1 × 10^-31 kg and moving with a velocity of 10⁷ ms^-1. Its de-Broglie wavelength will be;

_{λ =h/} mv ₌ 6.626 × 10^-34 kg m²s^-1  _{/ (} 9.1 × 10^-31 kg ×10⁷ ms^-1_{) = 0.727 × 10}-10_{m = 7.27 × 10}-11_m

This value of λ can be measured by the method similar to that for the determination of wave length of X-rays.

Let us now consider a ball of mass 10^-2 kg moving with a velocity of 10² ms^-1. Its de-Broglie wave length will be;

This value of λ can be measured by the method similar to that for the determination of wave length of X-rays.

Let us now consider a ball of mass 10^-2 kg moving with a velocity of 10² ms^-1. Its de-Broglie wave length will be;

_{λ =h/} mv ₌ 6.626 × 10^-34 kg m²s^-1  _{/ (}10^-2 kg ×10² ms^-11_{) = 6.62x10}-34_m

This wavelength is too small to be measured, and hence de-Broglie relation has no significance for such a large object.

Thus, de-Broglie concept is significant only for sub-microscopic objects in the range of atoms, molecules or smaller sub-atomic particles.