The Schrödinger Equation
The cornerstone of non-relativistic quantum mechanics — a wave equation describing how the quantum state of a system evolves over time.
Time-dependent form
$$i\hbar\,\frac{\partial}{\partial t}\Psi(\mathbf{r},t)
= \hat{H}\,\Psi(\mathbf{r},t)$$
Time-independent form
$$\hat{H}\,\psi(\mathbf{r}) = E\,\psi(\mathbf{r})$$
What the symbols mean
Formulated by Erwin Schrödinger in 1926, this equation plays the same role in quantum mechanics that Newton's second law plays in classical mechanics. It is a linear partial differential equation whose solution $\Psi$ contains all physically accessible information about the system.
$i$
Imaginary unit, $i^2 = -1$
$\hbar$
Reduced Planck constant $h/(2\pi) \approx 1.055 \times 10^{-34}\,\text{J·s}$
$\Psi$
Wave function — its squared modulus $|\Psi|^2$ gives the probability density
$\hat{H}$
Hamiltonian operator — represents the total energy of the system
$E$
Energy eigenvalue of the stationary state
As a bonus, the famous mass–energy equivalence: $E = mc^2$, where $c \approx 3 \times 10^8\,\text{m/s}$ is the speed of light.