Quantum Mechanical Model of Atom

2.6 Quantum Mechanical Model of Atom

Classical mechanics, built on Newton's laws, works beautifully for the world we can see — a ball rolling down a hill, planets orbiting the Sun. These are macroscopic objects with particle-like behaviour. But when we try to apply the same laws to electrons, atoms, and molecules, they fail completely. Two reasons stand out: classical mechanics ignores the dual behaviour of matter (that tiny particles also behave like waves) and it ignores the uncertainty principle.

The branch of science that embraces this dual behaviour is called quantum mechanics. It is a theoretical framework that studies the motion of microscopic objects — objects that show both wave-like and particle-like properties. It lays down the laws these objects obey. When you apply quantum mechanics to a macroscopic object (where wave properties are negligible), the results match classical mechanics perfectly. So quantum mechanics doesn't contradict classical physics — it contains it as a special case.

Quantum mechanics was developed independently in 1926 by Werner Heisenberg and Erwin Schrödinger. We will focus on the version based on wave motion, because that is the one that leads directly to the atomic model you need.

The Schrödinger Equation

The fundamental equation of quantum mechanics is the Schrödinger equation. Schrödinger won the Nobel Prize in Physics in 1933 for this work. The equation incorporates de Broglie's wave-particle duality, and solving it requires advanced mathematics. For a system whose energy does not change with time (like an atom or a molecule in a stable state), the equation is written as:

Here, $\hat{H}$ is a mathematical operator called the Hamiltonian. Schrödinger gave a recipe for constructing this operator from the total energy expression of the system. The total energy includes:

• the kinetic energies of all sub-atomic particles (electrons, nuclei)
• the attractive potential between electrons and nuclei
• the repulsive potential among electrons and among nuclei …