Education

(1)Physics

From physics for life sciences to quantum mechanics, electromagnetism, statistical physics, and solid state. Every course has its own demo: pick one.

Classically, a particle with less energy than the barrier always bounces back. The wave function leaks through anyway: the time-dependent Schrödinger equation, \(i\hbar\,\partial_t\psi = -\tfrac{\hbar^2}{2m}\partial_x^2\psi + V\psi\), solved live by the split-operator method (half a step of potential, a full step of kinetic energy in Fourier space, half a step of potential).

Each line follows \(\mathbf E = \sum_i k q_i\,\hat{\mathbf r}_i / r_i^2\) out of the positive charges, and the number of lines leaving a charge is proportional to its size, so their crowding shows the field’s strength (Gauss’s law).

Nothing here is random except the starting directions, yet elastic collisions alone spread the speeds onto the Maxwell–Boltzmann distribution (in two dimensions, \(f(v) \propto v\,e^{-mv^2/2kT}\)), with the temperature set by the energy the particles started with. Pushing the piston in does work on the gas: particles bounce off the moving wall faster than they hit it, so the whole distribution heats up, and pulling it back out cools it again.

(2)Mathematics and statistics

Calculus through complex analysis and abstract algebra, with differential equations and numerical methods on the applied side. Every course has its own demo: pick one.

Scrolling walks the matrix from the identity through a shear, a stretch, a rotation, and a reflection. Its columns (red and blue) are where the two basis vectors land, and your pointer is a vector \(\mathbf v\), with \(A\mathbf v\) in gold. The dashed lines are the eigenvectors, the directions the matrix only stretches, \(A\mathbf v = \lambda \mathbf v\). The shaded square’s area is scaled by the determinant (and flipped when it turns negative).

A frictionless spring, \(\ddot x = -x\), should go round the same circle forever. Euler’s method gains energy every step and spirals out; fourth-order Runge–Kutta, with error \(O(h^4)\) rather than \(O(h)\), stays on the circle even with large steps.

Each sample is drawn from the lopsided distribution on the left (an exponential). Their averages pile up on the right, and as the sample size grows the pile becomes the normal curve \(N(\mu, \sigma^2/n)\), whatever the shape you started from.

(3)Economics and finance

Micro and macro to the intermediate and advanced level, econometrics, game theory, and mathematical economics, plus finance and accounting. Every course has its own demo: pick one.

(4)Biology and chemistry

The first two years were life sciences: general chemistry, cell and molecular biology, genetics, biochemistry, physiology, and pharmacology. Every course has its own demo: pick one.