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

**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 = mc^{2} (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 = mc^{2}

But ν= c / λ

h . c / λ = mc^{2}

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 10^{7} ms^{-1}. Its de-Broglie wavelength will be;

_{λ} _{=h/ }mv
_{= }6.626 × 10^{-34}
kg m^{2}s^{-1}_{ / (}
9.1 × 10^{-31}
kg ×10^{7}
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 10^{2}
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 10^{2}
ms^{-1}. Its de-Broglie wave length will be;

_{λ} _{=h/ }mv
_{= }6.626 × 10^{-34}
kg m^{2}s^{-1}_{ / (}10^{-2}
kg ×10^{2}
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.

^{ }

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