Towards Quantum Mechanical Model of the Atom

Full Notes:

2.5.1 Dual Behaviour of Matter

Inspired by Einstein’s success in explaining the particle nature of light, Louis de Broglie proposed in 1924 that matter, like light, also exhibits wave-particle duality.

That is:

• Light, once thought to be a wave, also behaves like a stream of particles (photons).
• Particles such as electrons, though previously thought to be purely particles, must also exhibit wave properties.

De Broglie’s Hypothesis:

A moving material particle (like an electron) has a wavelength λ, given by:

 (Equation 2.22)

This is known as the de Broglie wavelength. Heavier objects have very small wavelengths – negligible for macroscopic bodies – but significant for electrons and sub-atomic particles.

This concept was later confirmed experimentally by Davisson and Germer in 1927, who showed that electrons can exhibit diffraction – a property of waves.

2.5.2 Heisenberg’s Uncertainty Principle

Werner Heisenberg (1927) formulated a principle that places fundamental limits on our ability to measure both the position and momentum simultaneously of microscopic particles.

Statement of the Principle:
“It is impossible to determine simultaneously, the exact position and exact momentum (or velocity) of a particle.”

 (Equation 2.23)

• Where: Δx = uncertainty in position; Δp = uncertainty in momentum (p = m × v); Δv = uncertainty in velocity; h = Planck’s constant

The product of uncertainties can never be zero. That is, the more precisely you know one quantity, the less precisely you can know the other.

Note that it applies to subatomic particles (e.g. electrons), not macroscopic objects. This is because when dealing with milligram-sized or heavier objects, the associated uncertainties are hardly of any real consequence.

Significance of the Uncertainty Principle

• It shows the limitation of classical mechanics in atomic-scale systems.
• According to this principle, we cannot assign definite orbits to electrons (as Bohr did), because both position and momentum cannot be known with certainty.
• Thus, it goes against the concept of well-defined orbits, making the Bohr model incompatible with quantum reality.
• Instead, quantum mechanics treats the electron as a wave spread over space, described by a probability distribution rather than a precise path.

Reasons for the Failure of the Bohr Model

Despite its success with hydrogen, the Bohr model could not survive under scrutiny from quantum developments. It failed due to the following reasons:

1. Wave-Particle Duality of Electrons:
   Bohr treated electrons purely as particles moving in fixed circular orbits. This ignored their wave-like properties (de Broglie’s theory), which are crucial at the atomic level.
2. Uncertainty Principle Violation:
   Bohr’s model assumes exact knowledge of electron position (in orbit) and momentum, which contradicts Heisenberg’s uncertainty principle.
3. Multi-Electron Systems:
   Bohr’s model could not explain atoms with more than one electron, including their spectral line complexities, electron repulsions and shielding, and fine and hyperfine spectral structure.
4. No Explanation for Zeeman and Stark Effects:
   The splitting of spectral lines in external magnetic or electric fields could not be explained.
5. No Basis in Wave Mechanics or Probability:
   The model lacked the conceptual and mathematical framework of quantum mechanics, particularly the Schrödinger equation, which replaced the idea of orbits with orbitals.

Conclusion

The ideas of de Broglie and Heisenberg marked a shift from classical orbits to quantum mechanics, where:

• Electrons are no longer seen as particles on defined paths.
• Instead, their behavior is described by probability functions and wave equations.

These developments laid the foundation for the quantum mechanical model of the atom.

Summary

• de Broglie proposed matter waves with wavelength λ = h/p.
• Heisenberg uncertainty shows limits on simultaneous position and momentum measurement.
• Definite orbits are replaced by probability-based orbitals.
• Bohr’s model fails for multi-electron atoms and modern quantum effects.