Quantum mechanical model of atom – Schrödinger Equation

Ordinary, everyday motion - a thrown ball, a moving car - is completely described by classical mechanics, built on Newton's laws. In classical mechanics, a particle's physical state at any instant is captured by its position and its momentum together; knowing both, along with the forces acting on it, lets you predict its entire future trajectory with certainty.

That whole framework breaks down for a microscopic particle like an electron, for a very specific reason: Heisenberg's uncertainty principle says position and momentum can never both be pinned down simultaneously with full precision, so the two ingredients classical mechanics depends on can never both be supplied at once. Classical mechanics also has no way of accounting for the wave-like character of matter, which becomes significant at these tiny scales. The combination of these two failures forced the development of an entirely new framework: quantum mechanics.

Schrödinger's wave equation. Erwin Schrödinger's central contribution was to express the electron's wave nature as a differential equation - one that describes how a mathematical quantity called the wavefunction, $\psi$, varies through space under the influence of whatever force field the electron is moving in. The version of this equation that does not involve time explicitly, appropriate for an electron in a stationary (unchanging) state, is called the time-independent Schrödinger equation:

Here $\hat{H}$ is called the Hamiltonian operator - a mathematical instruction combining the kinetic-energy part and the potential-energy part of the system - $\psi$ (the wavefunction) is a function of the particle's position coordinates, written $\psi(x,y,z)$, and $E$ is the total energy of the system. Explicitly, for a particle of mass $m$ moving in a potential $V$,

so that equation (2.12), written out in full, becomes

$\frac{\partial^2\psi}{\partial x^2}+\frac{\partial^2\psi}{\partial y^2}+\frac{\partial^2\psi}{\partial z^2}+\frac{8\pi^2m}{h^2}(E-V)\,\psi=0\qquad\text{— eq. (2.13)}$ …