the Schrodinger equation
/ SHROH-ding-er /
At the scale of atoms, a particle is not a tiny ball at a definite spot — it is a spread-out wave of possibility. The Schrodinger equation is the law that governs how that wave evolves in time. It is to quantum mechanics what Newton's law is to ordinary mechanics: give it the forces (a potential) and an initial state, and it tells you the state at every later moment. Lasers, transistors, MRI, and chemistry itself all rest on it.
The (time-dependent) equation is i hbar psi_t = -(hbar^2 / 2m) Laplacian psi + V(x) psi, where psi(x, t) is the complex-valued wavefunction, hbar is the reduced Planck constant, m the particle's mass, and V the potential energy. The quantity |psi|^2 is a probability density for finding the particle. Notice the i (square root of -1) in front: this is not the heat equation, despite the family resemblance (one time derivative, a Laplacian in space). That i makes it a dispersive equation — it conserves the total probability (the integral of |psi|^2) and does NOT smooth or dissipate; instead it spreads wave packets and produces interference. A standard solution method is separation of variables: seek psi(x,t) = phi(x) e^(-i E t / hbar); plugging in gives the time-independent equation -(hbar^2/2m) Laplacian phi + V phi = E phi, an eigenvalue problem whose allowed energies E are the quantized energy levels.
Solving that eigenvalue problem for the right V is most of introductory quantum mechanics: a harmonic-oscillator potential gives equally spaced energy levels and Hermite-function modes; the hydrogen atom (a Coulomb 1/r potential) is solved by separation in spherical coordinates and yields spherical harmonics and Laguerre functions — the orbitals of chemistry. The Schrodinger equation is the headline reason that the same PDE toolkit — separation of variables, eigenfunctions, special functions — that solves heat and wave problems also explains the periodic table.
Particle in a box (V = 0 on 0 < x < L, infinite walls): separation gives standing-wave modes phi_n(x) = sin(n pi x / L) with energies E_n proportional to n^2. These are exactly the Fourier sine modes of a vibrating string — the same spatial eigenvalue problem — but here the spacing of the energy levels is why atoms emit light at discrete colours.
Dispersive, not diffusive: the i conserves probability and creates interference.
Despite looking like the heat equation u_t = k Laplacian u, the Schrodinger equation does not smooth or decay — the factor of i changes everything, making it time-reversible and probability-conserving. Treating it as 'imaginary-time diffusion' is a useful trick but a different physical world.