TOWARDS QUANTUM MECHANICAL MODEL OF AN ATOM
Although Bohr’s model of the atom is widely recognized, scientists have continued to develop new and improved models. Two key concepts that have significantly contributed to the advancement of atomic models are the dual behavior of matter and the Heisenberg uncertainty principle.
Dual Behavior of Matter
In 1924, Louis de Broglie proposed that matter, like radiation, exhibits dual behavior—showing both wave-like and particle-like properties. This means that electrons, like photons, have both a wavelength and momentum. De Broglie derived an equation to relate the wavelength (λ) and momentum (p) of a material particle:
λ = h/mv = h/p
where h is Planck’s constant (6.626×10−34 Js), mmm is the mass of the particle, v is the velocity, and p is the momentum.
De Broglie’s prediction was experimentally verified when electrons were found to undergo diffraction, a property characteristic of waves. This discovery has been utilized in constructing electron microscopes. Just as a conventional microscope uses the wave nature of light, an electron microscope uses the wave-like behavior of electrons.
An electron microscope can achieve magnification up to 15 million times, making it a powerful tool in scientific research. According to de Broglie, every moving object has a wave-like character, but the wave properties of ordinary objects are difficult to detect due to their very short wavelengths.
Experimental detection of wavelengths associated with electrons and other sub-atomic particles is possible. For example, to find the wavelength of a ball with a mass of 0.1 kg moving at a velocity of 10 m/s, we use de Broglie’s equation:
λ = h/mv = (6.626 x 10-34)/ (0.1 x 10) = 6.626 x 10-34m
Heisenberg’s Uncertainty Principle
In 1927, Werner Heisenberg formulated the uncertainty principle, influenced by the concepts of the dual behavior of matter and radiation.
Heisenberg’s principle states that it is impossible to determine simultaneously the exact position and exact momentum (or velocity) of an electron. The principle can be expressed mathematically as:
∆x × ∆px≥ h/4π OR
∆x × ∆(mvx) ≥h/4π OR
∆x ×∆vx≥ h/4πm
where Δx is the uncertainty in position and ∆px or Δvx is the uncertainty in momentum (or velocity) of the particle. If Δx is small, then Δvx is large, and vice versa. Therefore, if we physically measure an electron’s position or velocity, the result is always uncertain or fuzzy.
Example
To explain the uncertainty principle, imagine measuring the thickness of a sheet of paper with an unmarked ruler. The results would be inaccurate. To measure correctly, you would need a ruler with units smaller than the thickness of the paper. Similarly, to measure the position of an electron, we need a tool with a resolution smaller than the electron's dimensions. We can observe an electron by illuminating it with light or electromagnetic radiation. This light should have a wavelength smaller than the dimensions of the electron.
The collision of high-momentum photons (with p=h/λ ) with electrons changes the electrons' energy, allowing us to calculate the position but not the velocity.
Significance of the Uncertainty Principle
The uncertainty principle eliminates the possibility of defining exact paths or trajectories for electrons and similar particles. The trajectory of an object involves knowing its location and velocity at a given instant and predicting its future position. Since it is impossible to determine the exact position and velocity of an electron simultaneously, we cannot predict its exact trajectory.
The uncertainty principle is significant only for microscopic objects, and negligible for macroscopic objects. For instance, applying the uncertainty principle to an object with a mass of 1 milligram (10^-6 kg):
∆v.∆x = h/4πm = 6.626 x10-34Js/(4 x 3.1416 x 10-6kg)≈ 10-28m2s-1
For objects of this size or larger, the uncertainties are insignificant. However, for an electron with a mass of 9.11 x 10^-31 kg:
∆v.∆x = h/4πm =6.626 x10-34Js/(4 x 3.1416 x 9.11 x 10-31kg) = 10-4m2s-1
If we try to determine an electron's position with an uncertainty of about 10-8 m, the uncertainty in velocity (Δv) would be:
10-4m2s-1/10-8m≈ 104ms-1
This large uncertainty in velocity invalidates the concept of electrons moving in precise orbits as proposed by Bohr. Therefore, in the quantum mechanical model of the atom, the precise statements of the position and momentum of electrons are replaced by statements of probability.