# Get the Knowledge that sets you free...Science and Math for K8 to K12 students

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## Wave Nature of Particles

Are you a particle or wave? Schrodinger's equation grew out of the idea that particles such as electrons behave like particles in some situations and like waves in others: that's the so–called wave – particle duality, One question that comes up immediately is why we never see big objects like tables, chairs, or ourselves behave like waves? Lets see what this topic has got to clear these questions !

## After completing the topic, the student will be able to:

• Understand and relate the wave nature of particles using de Broglie wavelength to the observations made in electron diffraction tube.
• Explore Davisson–Germer experiment in finding the wavelength of a moving electron in a crystal.
• Determine the wave–function probability of an electron in a particular shell or orbit using Schrodinger wave equation.
• Investigate and conclude that it is not possible to determine both the position and momentum of an electron in a single measurement; as discussed by Hisenberg in his uncertainty rule.
• Probe about the non classical degree of freedom possessed by an electron and discuss the dirac equation which relates Quantum theory of matter and Quantum theory of radiation.
de Broglie wavelength Radiolarian is a protozoa viewed under light microscope and electron microscope. The high-speed electrons have wavelengths much smaller than the wavelength of the visible light, hence the resolution of the image is more in case of electron microscope.
De Broglie's explanation of Bohr's second postulate of quantisation

Light is a form of energy which sometimes behaves as waves and sometimes as particles(photons). Matter also can behave both like particles as well as waves. A wave is specified by the quantities like frequency , wavelength λ, amplitude and intensity. A particle is specified by its mass m, velocity v, momentum p and energy E.

The concepts like interference, diffraction and polarization tell us that light is a wave. Experiments like photo–electric effect, Compton effect, black-body radiations, X–ray spectra shows light in its particle nature. Louis de Broglie (1892–1987), a French physicist was the first to draw attention to this possibility.

More specifically, he said that if a particle of mass m, moves with a velocity v then it behaves like a wave having a wavelength λ given by λ =h/(mv). He received a Nobel prize in 1928. Such a matter wave is sometimes referred to as de Broglie wave, and λ as the de Broglie wavelength. The wave associated with a moving particle is called matter wave or de Broglie wave and controls the particle in every respect.

The intensity of a matter wave at a point represents the probability of the associated particle (e.g. electron) being there. Therefore, if the intensity of matter wave is large in a certain region, there is a greater probability of the particle being found there.

Electron diffraction pattern Demonstration of wave–particle duality. An electron gun has been fired at a thin sheet of graphite. The electrons passed through and hit a luminescent screen, producing the patterns of rings associated with diffraction. Diffraction occurs when a wave passes through an aperture similar in size to its wavelength. But electrons are particles, so should not exhibit the same phenomenon unless they can also behave like waves. de Broglie correctly deduced that this was the case and that particles have wavelengths inversely proportional to their momentum.

According to Planck's quantum theory, the energy of a photon of radiation of frequency f and wavelength λ is given by: E = hf ............(i)

where h = Planck's constant,

If photon is considered as a particle of mass m, then according to Einstein's energy–mass relation, the energy E of the photon is given by: E = mc2 ..............(ii)

For eq's. (i) and (ii) we have, hf = mc2

The quantity mc, is the momentum p of the photon having mass m and travelling with velocity c.

Eq. (iii) gives de Broglie wavelength for a photon. According to de Broglie eq. (iii) is applicable to both the photons of radiation and other material particles. Thus if a material particle has mass m and moves with a velocity v, its momentum is p = mv. According to de Broglie, the wavelength λ of the wave associated with this moving particle is

Eq.(iv) is de Broglie wave equation for a moving material particle.