Heisenberg’s Uncertainty Principle and Quantum Mechanical Model

Heisenberg’s Uncertainty Principle

According to this principle, it is impossible to determine the position and momentum of a small microscopic moving particle like an electron with absolute accuracy or certainty.

According to him, the uncertainty in position (Δx) and uncertainty in momentum (Δp = m.Δv) is equal to or greater than h/4π.

Mathematically,

Δx. Δp ≥ h/4π

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 the macroscopic moving particle.

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

Quantum Mechanical Model of an atom

In 1926, Ervin Schrodinger developed an atomic model based on the wave and particle nature of electron which is known as the quantum mechanical model of the atom.

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

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.

The significance of Wave Function 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.

The significance of Ψ²:

It is probability factor. It describes the probability of finding an electron in a space. Space, where the probability of finding an electron is maximum, is described as an 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 the nucleus.