Heisenberg Uncertainty Principle

All moving objects that we see around us e.g., a car, a ball thrown in the air etc., move along definite paths. Hence their position and velocity can be measured accurately at any instant of time. Is it possible for subatomic particle also?

As a consequence of  dual nature of matter, Heisenberg, in 1927 gave a principle about the uncertainties in simultaneous measurement of position and momentum (mass × velocity) of small particles.

This principle states

“It is impossible to measure simultaneously the position and momentum of a small microscopic moving particle with absolute accuracy or certainty” i.e., if an attempt is made to measure any one of these two quantities with higher accuracy, the other becomes less accurate.

The product of the uncertainty in position (Δx) and the uncertainty in the momentum (Δp = m.Δv where  m is the mass of the particle and Δv is the uncertainty in velocity) is equal to or greater than h/4π where h is the Planck’s constant.

Thus, the mathematical  expression for the Heisenberg’s uncertainty principle is simply written as

$\large \Delta x . \Delta p \ge \frac{h}{4\pi}$

Explanation of Heisenberg’s uncertainty principle:

Suppose we attempt to measure both the position and momentum of an electron, to pinpoint the position of the electron we have to use light so that the photon of light strikes the electron and the reflected photon is seen in the microscope.

As a result of the hitting, the position as well as the velocity of the electron are disturbed. The accuracy with which the position of the particle can be measured depends upon the wavelength of the light used. The uncertainty in position is ± λ .

The shorter the wavelength, the greater is the accuracy. But shorter wavelength means higher frequency and hence higher energy. This high energy photon on striking the electron changes its speed as well as direction. But this is not true for macroscopic moving particle.

Hence Heisenberg’s uncertainty principle is not applicable to macroscopic particles

Question : Why electron cannot exist inside the nucleus according to Heisenberg’s uncertainty principle ?

Sol: Diameter of the atomic nucleus is of the order of 10–15 m

The maximum uncertainty in the position of electron is 10–15 m

Mass of electron = 9.1 × 10–31 kg

$\large \Delta x . \Delta p = \frac{h}{4\pi}$

$\large \Delta x . ( m \Delta v ) = \frac{h}{4\pi}$

$\large \Delta v = \frac{h}{4\pi} . \frac{1}{\Delta x m}$

$\large \Delta v = \frac{6.63 \times 10^{-34}}{4 \times 3.14} . \frac{1}{10^{-15} \times 9.1 \times 10^{-31}}$

Δv = 5.80 × 1010 ms–1

This value is much higher than the velocity of light and hence not possible.

Quantum Mechanical Model of atom:

The atomic model which is based on the particle and wave nature of the electron is known as wave or quantum mechanical model of the atom.

This was developed by Ervin Schrodinger in 1926. This model describes the electron as a three dimensioinal wave in the electronic field of positively charged nucleus.

Schrodinger derived an equation which describes wave motion of an electron. The differential equation is

$\Large \frac{d^2 \psi}{dx^2} + \frac{d^2 \psi}{dy^2} + \frac{d^2 \psi}{dz^2} + \frac{8\pi^2 m}{h^2} (E-V)\psi = 0$

where x, y, z are certain coordinates of the electron, m = mass of the electron E = total energy of the electron. V = potential energy of the electron; h = planck’s constant and Ψ (psi) = wave function of the electron.

Significance of Ψ :The wave function may be regarded as the amplitude function expressed in terms of  coordinates x, y and z. The wave function may have positive or negative values depending upon the value of coordinates.

The main aim of Schrodinger equation is to give solution for probability approach. When the equation is solved, it is observed that for some regions of space the value of Ψ is negative.

But the probability must be always positive and cannot be negative, it is thus, proper to use Ψ2 in favour of Ψ .

Significance of Ψ2: Ψ2 is a probability factor. It describes the probability of finding an electron within a small space. The space in which there is maximum probability of finding an electron is termed as orbital.

The important point of the solution of the wave equation is that it provides a set of numbers called quantum numbers which describe energies of the electron in atoms, information about the shapes and orientations of the most probable distribution of electrons around nucleus.