Heisenberg Uncertainty Principle
Position and momentum (or velocity) obey the Heisenberg Uncertainty Principle: x and p are Fourier-conjugate, meaning they cannot be defined simultaneously with arbitrary precision. The uncertainty is very small because Planck's constant is very small, so it does not show up in the macroworld, only when objects are molecule-sized.
Pauli Exclusion Principle
The Pauli Exclusion Principle states that particles of matter are not gregarious; they avoid one another on x-p plots. We can combine Heisenberg's and Pauli's principles to fill electrons into energy levels: quantum mechanics allows only certain energies when a particle is confined (such as an electron in a molecular "box").
E₋n = h²n²/8mL²
Photons, particles of light energy, are gregarious and have energy E=hν (Planck's Law). They can be absorbed or emitted by a molecule, raising or lowering the energy of the molecule to a different energy level.
Schrödinger realized that one of the two Fourier-conjugate variables (usually p) can be expressed in terms of the other (usually x), yielding the Schrödinger Equation. This equation allows us to solve for wave functions Ψ(x), whose magnitude-squared tells us the probability of finding an electron at x. Wave functions can oscillate and have negative or positive lobes.
The magnitude-squareds of wave functions Ψ(x) can tell the probability of a particle being in a particular locale. When graphed on a 3D graph, the resulting shapes are that of atomic orbitals.
Sums of Wave Functions
Sums of Wave Functions that solve Schrödinger's equations are also valid wave functions. Adding wave functions can either increase or cancel the probability of finding a particle at a position x. This can shift electrons in between nuclei (creating chemical bonds), or away (allowing the nuclei to repel one another and breaking a bond).