Quantum Mechanics Formulas

Quantum mechanics describes things so small that light comes in lumps and particles behave like waves. The formulas are shorter than their reputation. Each page here says what the symbols mean, what the formula predicts and what it does not say.

7 formulas, in the order that makes them easiest to learn:

Planck-Einstein Relation (E = hf)

$$E = h f$$

E = hf says each packet of light carries energy in proportion to its frequency: higher frequency means more energy per packet, and brightness does not change it.

Photoelectric Effect

$$K_{\max} = h f - \phi$$

Light can knock electrons out of a metal, but only if each packet of light carries enough energy, and the leftover energy becomes the electron’s speed.

De Broglie Wavelength

$$\lambda = \frac{h}{p} = \frac{h}{m v}$$

Every moving object has a wavelength, and the heavier or faster it moves, the shorter that wavelength gets.

Heisenberg Uncertainty Principle

$$\Delta x \, \Delta p \ge \frac{\hbar}{2}$$

The more tightly a particle’s position is pinned down, the more its speed must spread out, and no instrument can beat that trade-off.

Born Rule

$$P(x) = |\psi(x)|^2$$

The Born rule says the chance of finding a particle at a spot, or of getting an outcome, is the squared size of its wave at that spot.

Schrödinger Equation

$$i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi$$

The Schrödinger equation says how a quantum particle’s wave changes over time, the way Newton’s laws say how a moving ball changes its position.

Hydrogen Energy Levels

$$E_n = -\frac{13.6\ \text{eV}}{n^2}$$

A hydrogen electron can only sit on certain energy steps, and every jump between steps gives off or takes in light of one exact color.

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