Heinsberg Uncertainty Principle

In the experiments for the determination of specific charge of an electron, it was shown that the electrons get deflected by the application of electric and magnetic fields. It means that the electron behave as a particle and it has definite mass, momentum and energy which can be measured with desired accuracy. But the experiments of Davidson and Germer, and Thomson showed clearly the wave nature of electrons and the wavelength of the electron is given.


Here λ is the Planck’s constant and mv is the momentum of the electron. A wave extends through space and it is difficult to locate the exact position of an electron behaving as a wave, at any given instant of time.

The dual behaviour of electron as a particle and as a wave presents difficulty in locating the exact position and momentum at the same time. The difficulty is overcome according to Heisenberg’s uncertainty principle.

The principle states that the exact position and momentum of a particle (say electron) cannot be determined simultaneously with a desired accuracy. Taking Δ x as the error in determining its position and Δp the error in determining its momentum at the same instant, these quantities are related as follows:



The product of the two errors is approximately of the order of Planck’s constant.

Multiplying and dividing the left hand side of equation (1) by v, the velocity of the particle



Here, ΔE represents the uncertainty in the measurement of the energy of a particle and Δt the uncertainty in the measurement of time.

In equation (1), the product Δx Δp is of the order of h/2π. If Δx is small, Δp will be large and vice versa. It means that if one quantity is measured accurately, the other quantity becomes less accurate. A similar argument holds goods for ΔE Δt also.

Consider a beam of electrons travelling in the direction shown in fig. The slit AB of width Δx is perpendicular to the path of the electrons. Before entering the slit, the electron has a definite momentum p = mv. After passing through the slit the electron gets diffracted and acquires a momentum along OG. The angular deflection θ depends upon the component of momentum parallel to the slit, i.e. Δp = p sin θ = pθ for small angular deflection. The angle θ₀ corresponding to the direction of first minimum of the diffraction pattern is


But, pλ = h





∴ Δp Δx ~ h

The probable deflection θ of the electron is less than θ₀ and according to Heisenberg, the uncertainty relationship is,



In general, any experiment that measures position or momentum accurately, introduces an uncertainty in the other. The uncertainty is never less than the predicted value according to Heisenberg’s relation. Thus, any instrument cannot measure the quantities more accurately than predicted by Heisenberg’s principle of uncertainty or indeterminacy.

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