# Mathematical Expression of Uncertainty Principle

A careful analysis of limitations involved in the simultaneous determination of position and momentum of a subatomic particle leads to the following statement of Heisenberg's uncertainty principle.

It is not possible to design an experiment, with the help of which one can determine simultaneously the precise values of both position and momentum of subatomic particles. The mathematical expression of uncertainty principle is

Î”x.Î”p h/4

where Î” p is the uncertainty in the momentum, Î”x is the uncertainty in position and h is Planck's constant. The sign means that the product of uncertainties in p and x is equal to or greater than h.4 . It is never less than this. Obviously, if we make Î” p small, Î” x has a large value and vice versa.

**Characteristics of Electromagnetic Radiations**

A sub-atomic particle (such as electron, proton, neutron, hydrogen atom, etc.) besides having the particle nature also has the significant wave nature. Classical Newton's law of motion cannot describe its mechanics. The qualitative explanation, as proposed by Werner Heisenberg, may be understood from the following analysis.

**Limitations Involved in the Simultaneously Determination of Position and Momentum of a Particle**

An object is visible when the photon of light after colliding with the particle reaches the human eyes. Moreover, from the principles of optics we know that if the particle is to be located within a distance , then we must use light with a wavelength at least that small. Now, to locate an electron, which is a very tiny particle, we would have to use radiation of very small wavelength (of the order of the size of electron). No doubt, we can determine the position of an electron very precisely by choosing a light of very small wavelength. Now, according to de Broglie's relation, each photon would carry an exceedingly large value of momentum. During the process of locating the electron, some momentum of the photon would be transferred to the electron during the collision of the two. Hence, the momentum of the electron would become uncertain by an amount equal to the momentum transferred. Now, if we wish to determine the momentum of an electron precisely, we would have to use the photon of smaller momentum so that during collision,the transfer of momentum is infinitesimally small. But this will make the position of the electron more uncertain as the wavelength of radiation employed would be large.