## Learning objectives

## 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.

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.

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 = mc^{2} ..............(ii)

For eq's. (i) and (ii) we have, hf = mc^{2}

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.