Wave Nature of Particle - the de Broglie Hypothesis( College Level)

In experiments like photoelectric effect and Compton effect, radiation behaves like particles. de Broglie, a french physicist asked whether in some situations, the reverse could be true, i.e., would objects which are generally regarded as particles (e.g. electrons) behave like waves ? In 1924 de Broglie postulated that we can associate a wave with every material object. In analogy with photons, he proposed that the wavelength associated with such a matter wave is related to the particle momentum through the relationship



where is the Planck's constant

Wavelike behaviour of a macroscopic object is difficult to detect as the wavelength is very small.
However, wave nature of particles may be detected in diffraction experiments where the dimensions of the obstacles are comparable with the wavelength of matter wave incident on the obstacle.



Example :Electron Diffraction from a Crystal

Consider a beam of electron with a speed m/s corresponding to a wavelength nm. Such a wave may be diffracted by gratings with separation of similar order as that of the wavelength. Crystals provide such natural gratings.
Davisson - Germer Experiment :
Experimental confirmation of de Broglie hypothesis was provided in 1926 by Davisson and Germer, who studied diffraction of a beam of electrons from the surface of a nickel crystal.



















A beam of electrons from a heated filament, accelerated through a potential difference is made to strike the surface of a crystal of Ni. Electrons are scattered in all directions and may be detected by an array of detectors located at various angles of scattering. It is found that the intensity of scattered beam is maximum at some particular angles of incidence, in the same manner as the case when a beam of x-rays strikes the crystal
In Davisson - Germer experiment, the electron beam was accelerated through a potential difference of volts. The kinetic energy of the electron is thus 54 eV. The de Broglie wavelength associated with an electron accelerated through a potential difference may be expressed as


Notes:

The dual nature not only is exhibited by radiation but is also associated with matter. In some experiments matter shows wave character.
de Broglie hypothesis poastulates a wavelength of with a particle having a momentum .
 Experimental confirmation of wave nature of matter comes from experiments such as Davisson Germer experiments
on electron diffraction from crystals. It is seen that the intensity of scattered beam is maximum at those points where one would expect Laue spots in x-ray diffraction assuming the electronsare waves with de Broglie wavelength.
 Bohr model can be understood by postulating that stable orbits in atoms are those which are standing waves of
electrons.
One can perform double slit experiment with electrons, similar to the way Young's double slit experiment is
performed with light waves. The intensity pattern obtained on a screen is very similar in both cases.
 According to the principle of complementarity one cannot obtain information on both the wave nature and particle
nature of matter or radiation in the same experiment.
Heisenberg uncertainty principle states that one cannot precisely measure both position and momentum of a
particle in the same experiment.


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