According to de-Broglie, all material particles in motion possess wave characters. Therefore, the wave-length associated with a particle of mass ‘m’ moving with a velocity ‘v’ is given by the relation-
λ = h/mv
Where, h = Planck’s constant
λ = Wave length,
mv = Momentum of the particle.
Derivation of de-Broglie’s Equation:
Let us consider a photon of mass ‘m’ having energy ‘E’ moving with a velocity ‘C’ i.e. velocity of light. Then according to Planck’s quantum theory of radiation,
E = hυ ——– (i)
Where, h = Planck’s constant
υ = Frequency of the radiation.
Again from Einstein mass and energy relationship,
E = mc2 ——– (ii) Where, m = Mass and
c = Velocity of light.
Combining of relation (i) and (ii), we get,
hυ = mc2 υ = c/ λ
or h.c/λ = mc2
or h/λ = mc
Therefore, λ = h/mc
de- Broglie Pointed out that the same equation might be applied to a material particle by using ‘m’ for the mass of the particle instead of the mass of photon and replacing ‘c’, the velocity of photon by ‘v’, the velocity of the particle. Thus,
λ = h/mv
Since, mv = P, the momentum of the particle, the above equation becomes,
λ = h/p = Planck’s constant/momentum
Since, h is a constant, then,
Wavelength, λ ∞ 1/ momentum
It means that the wave length of a particle in motion is inversely proportional to its momentum. These waves are different from the electromagnetic waves.
Significance of de- Broglie’s wave:
The wave character puts some restriction on how precisely we can express the position of an electron or any other small moving particle. This is due to the fact that unlike particles, wave does not occupy a well defined position in space and are delocalized. The wave nature of matter, however, has no significance for the objects of ordinary size because wavelength of the wave associated with them is too small to be detected. This can be explain by the following examples,
1) Considering an electron of mass 9.1×10-31kg, moving with a velocity 107ms-1. The de-Broglie wave length will be 7.7×10-11
2) Considering a ball of mass 10-2kg, moving with a velocity 102ms-1. Its de-Broglie wave length will be 6.62×10-24
This wave length is very small to be measured. Hence de-Broglie’s relation has no significance for such a large objects. de-Broglie’s relation is significant only for sub-microscopic objects. e.g. atom, molecules, etc.
Justification for the dual nature of electron: The particle and wave nature of electron can be justified on the basis of the following observations:
1) Particle nature of electron: An electron exhibits all the characters of a particle i.e. it has mass, energy, momentum and charge. When an electron is made to fall on a screen coated with zinc sulphide, it produces a spot of light known as scintillation. It has been observed that one electron produce only one scintillation point. It means that the scintillation and as such the striking electron must be localized and not spread out like wave. In other wards, an electron behaves like a particle.
2) Wave nature of electron: de- Broglie’s concept of wave nature of electron was experimentally verified by Davission and Germer in 1927. In this experiment, a beam of electrons obtained from a heated tungsten filament is accelerated by using a high positive potential. When this fine beam of accelerated electrons is allowed to fall on a large single crystal of nickel, the electrons are scattered from the crystal in different direction. The diffracted pattern thus obtained is similar to the diffracted pattern of X-rays.
Since X-ray has wave character, therefore, the electrons must also have wave character associated with them. From the above discussion, it is clear that an electron behaves both as particle as well as wave i.e. it has dual character.
N.B.: de-Broglie’s concept can be applied not only electron but also to other particles like neutrons, protons, atoms and molecules.
Limitation of de-Broglie’s concept:
de-Broglie’s concept applies only to particles in a force free environment. Thus, it cannot be applied directly to an electron in an atom, where the electron is subjected to the attractive forces of the nucleus.
Difference between particle and wave:
Let us try to understand clearly the difference between these two terms by the following points of distinction between particle and wave-
Particle Wave
i) A particle occupies a well defined position i) A wave is spread out in space i.e. it is
in space i.e. it is localized in space. delocalized in space.
ii) Two particles cannot be simultaneously ii) Two or more waves can co-exist in the same
occupy the same position in space. same region of space.
iii) When a number of particles are present iii) When a number of wave are present in a given
in a given region of space, then their sum region of space, the amplitude of the resultant
is equal to the number of individual part- wave can be either smaller or larger than that
icles. In other wards, two particles do not of the individual wave. It is due to interference
interfere.