# 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.

Page: https://formulabro.yunusemrevurgun.com/quantum-mechanics/  
Updated: 2026-10-22

## [Planck-Einstein Relation (E = hf)](https://formulabro.yunusemrevurgun.com/quantum-mechanics/planck-einstein-relation/)

$$
E = h f
$$

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.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/planck-einstein-relation.md

## [Photoelectric Effect](https://formulabro.yunusemrevurgun.com/quantum-mechanics/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.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/photoelectric-effect.md

## [De Broglie Wavelength](https://formulabro.yunusemrevurgun.com/quantum-mechanics/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.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/de-broglie-wavelength.md

## [Heisenberg Uncertainty Principle](https://formulabro.yunusemrevurgun.com/quantum-mechanics/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.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/heisenberg-uncertainty-principle.md

## [Born Rule](https://formulabro.yunusemrevurgun.com/quantum-mechanics/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.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/born-rule.md

## [Schrödinger Equation](https://formulabro.yunusemrevurgun.com/quantum-mechanics/schrodinger-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.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/schrodinger-equation.md

## [Hydrogen Energy Levels](https://formulabro.yunusemrevurgun.com/quantum-mechanics/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.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/hydrogen-energy-levels.md

## [Photon Momentum](https://formulabro.yunusemrevurgun.com/quantum-mechanics/photon-momentum/)

$$
p = \frac{h}{\lambda} = \frac{E}{c}
$$

A photon has no mass, yet it carries momentum, so light can push on objects, and the shorter its wavelength, the harder each photon pushes.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/photon-momentum.md

## [Compton Scattering](https://formulabro.yunusemrevurgun.com/quantum-mechanics/compton-scattering/)

$$
\lambda' - \lambda = \frac{h}{m_e c}\left(1 - \cos\theta\right)
$$

When an X-ray bounces off an electron it loses some energy, so its wavelength gets longer, and the amount depends only on the angle it bounces through.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/compton-scattering.md

## [Planck's Law](https://formulabro.yunusemrevurgun.com/quantum-mechanics/plancks-law/)

$$
B_\lambda(\lambda, T) = \frac{2 h c^2}{\lambda^5} \cdot \frac{1}{e^{h c / (\lambda k_B T)} - 1}
$$

Planck's law says how bright a hot object glows at each wavelength, and the mix of colors depends on its temperature.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/plancks-law.md

## [Wien's Displacement Law](https://formulabro.yunusemrevurgun.com/quantum-mechanics/wiens-displacement-law/)

$$
\lambda_{\max} = \frac{b}{T}
$$

Wien's law says a hotter object is brightest at a shorter wavelength, and the peak wavelength is a fixed constant divided by the temperature.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/wiens-displacement-law.md

## [Stefan-Boltzmann Law](https://formulabro.yunusemrevurgun.com/quantum-mechanics/stefan-boltzmann-law/)

$$
P = \varepsilon \sigma A T^4
$$

The Stefan-Boltzmann law says a hot surface gives off radiated power that grows with the fourth power of its absolute temperature.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/stefan-boltzmann-law.md

## [Rydberg Formula](https://formulabro.yunusemrevurgun.com/quantum-mechanics/rydberg-formula/)

$$
\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)
$$

The Rydberg formula predicts the exact color of light a hydrogen atom gives off when its electron drops from one energy level to a lower one.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/rydberg-formula.md

## [Bohr Radius](https://formulabro.yunusemrevurgun.com/quantum-mechanics/bohr-radius/)

$$
a_0 = \frac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}
$$

The Bohr radius is the natural size of a hydrogen atom, about 0.0529 nanometers, built from fixed physical constants.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/bohr-radius.md

## [Hydrogen-Like Atom Energy](https://formulabro.yunusemrevurgun.com/quantum-mechanics/hydrogen-like-atom-energy/)

$$
E_n = -\frac{Z^2}{n^2} \times 13.605693\ \text{eV}
$$

An atom or ion with only one electron has energy levels that follow the hydrogen pattern, scaled up by the square of its nuclear charge.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/hydrogen-like-atom-energy.md

## [Electron Shell Capacity](https://formulabro.yunusemrevurgun.com/quantum-mechanics/electron-shell-capacity/)

$$
N_{\max} = 2 n^2
$$

The shell with number n can hold at most twice n squared electrons, because it has n squared places and each place takes two electrons.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/electron-shell-capacity.md

## [Particle in a Box](https://formulabro.yunusemrevurgun.com/quantum-mechanics/particle-in-a-box/)

$$
E_n = \frac{n^2 h^2}{8 m L^2}, \qquad n = 1, 2, 3, \dots
$$

A particle trapped between two walls can only have certain energies, and this formula gives them: the narrower the box, the higher the energies.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/particle-in-a-box.md

## [Quantum Harmonic Oscillator](https://formulabro.yunusemrevurgun.com/quantum-mechanics/quantum-harmonic-oscillator/)

$$
E_n = \hbar \omega \left(n + \frac{1}{2}\right), \qquad n = 0, 1, 2, \dots
$$

A quantum particle on a spring can only have energies in equal steps, and even its lowest step sits above zero.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/quantum-harmonic-oscillator.md

## [Quantum Tunneling Probability](https://formulabro.yunusemrevurgun.com/quantum-mechanics/quantum-tunneling-probability/)

$$
T \approx e^{-2\kappa L}
$$

A quantum particle can sometimes pass through a barrier that is too high for it to climb over, and this formula estimates how often.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/quantum-tunneling-probability.md

## [Wave Function Normalization](https://formulabro.yunusemrevurgun.com/quantum-mechanics/wave-function-normalization/)

$$
\int_{-\infty}^{\infty} |\psi(x)|^2\, dx = 1
$$

Normalizing a wave function means scaling it so the chances of finding the particle, added over every place, total exactly one.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/wave-function-normalization.md

## [Expectation Value](https://formulabro.yunusemrevurgun.com/quantum-mechanics/expectation-value/)

$$
\langle x \rangle = \int_{-\infty}^{\infty} x\, |\psi(x)|^2\, dx
$$

The expectation value is the average result you would get if you repeated a measurement many times on identical particles.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/expectation-value.md

## [Qubit State](https://formulabro.yunusemrevurgun.com/quantum-mechanics/qubit-state/)

$$
|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \qquad |\alpha|^2 + |\beta|^2 = 1
$$

A qubit is a quantum bit that can hold a mix of 0 and 1, described by two amplitudes whose squared sizes give the chances of each result.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/qubit-state.md

## [Bloch Sphere](https://formulabro.yunusemrevurgun.com/quantum-mechanics/bloch-sphere/)

$$
|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\varphi}\sin\frac{\theta}{2}|1\rangle, \qquad P(0) = \cos^2\frac{\theta}{2}
$$

The Bloch sphere is a ball whose points are all the possible single-qubit states, with the top and bottom poles as 0 and 1.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/bloch-sphere.md

## [Stationary State Time Evolution](https://formulabro.yunusemrevurgun.com/quantum-mechanics/stationary-state-time-evolution/)

$$
\Psi(x, t) = \psi(x)\, e^{-i E t/\hbar}, \qquad |\Psi(x, t)|^2 = |\psi(x)|^2
$$

A stationary state has one fixed energy, and its chances of finding the particle stay the same even though its phase keeps turning.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/stationary-state-time-evolution.md

## [Energy-Time Uncertainty Relation](https://formulabro.yunusemrevurgun.com/quantum-mechanics/energy-time-uncertainty/)

$$
\Delta E \, \Delta t \gtrsim \frac{\hbar}{2}
$$

A state that lasts only a short time cannot have one sharp energy, so the light it gives off has a blurred color.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/energy-time-uncertainty.md

## [Canonical Commutation Relation](https://formulabro.yunusemrevurgun.com/quantum-mechanics/canonical-commutation-relation/)

$$
[\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = i\hbar
$$

In quantum mechanics the order of measuring position and momentum changes the result, and that mismatch is the root of the uncertainty trade-off.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/canonical-commutation-relation.md

## [Orbital Angular Momentum Quantization](https://formulabro.yunusemrevurgun.com/quantum-mechanics/angular-momentum-quantization/)

$$
|\mathbf{L}| = \sqrt{\ell(\ell + 1)}\,\hbar \qquad L_z = m_\ell\,\hbar \qquad m_\ell = -\ell, \ldots, \ell
$$

An electron's orbital motion around a nucleus can only take certain fixed sizes and directions, like steps on a staircase.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/angular-momentum-quantization.md

## [Spin One-Half](https://formulabro.yunusemrevurgun.com/quantum-mechanics/spin-one-half/)

$$
S_z = m_s\,\hbar, \qquad m_s = \pm\frac{1}{2}, \qquad |\mathbf{S}| = \frac{\sqrt{3}}{2}\,\hbar
$$

An electron has a built-in angular momentum that can be measured along any direction as only one of two values, up or down.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/spin-one-half.md

## [Zeeman Effect](https://formulabro.yunusemrevurgun.com/quantum-mechanics/zeeman-effect/)

$$
\Delta E = m_\ell\,\mu_B\,B
$$

A magnetic field shifts an atom's energy levels by small equal steps, so one spectral line splits into several.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/zeeman-effect.md

## [Fine-Structure Constant](https://formulabro.yunusemrevurgun.com/quantum-mechanics/fine-structure-constant/)

$$
\alpha = \frac{e^2}{4\pi\varepsilon_0\,\hbar c} \approx \frac{1}{137.036}
$$

The fine-structure constant is a pure number that sets how strongly charged particles and light interact, and its value is close to 1 over 137.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/fine-structure-constant.md

## [Double-Slit Interference](https://formulabro.yunusemrevurgun.com/quantum-mechanics/double-slit-interference/)

$$
d \sin\theta = m\lambda
$$

Waves from two narrow openings overlap, and at certain places they add into bright bands while at others they cancel into dark ones.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/double-slit-interference.md

## [Bragg's Law](https://formulabro.yunusemrevurgun.com/quantum-mechanics/bragg-law/)

$$
n\lambda = 2d\sin\theta
$$

Waves bounce off the evenly spaced layers of a crystal, and they add into a strong reflection only at certain angles.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/bragg-law.md

## [Thermal de Broglie Wavelength](https://formulabro.yunusemrevurgun.com/quantum-mechanics/thermal-de-broglie-wavelength/)

$$
\lambda_{\text{th}} = \frac{h}{\sqrt{2\pi m k_B T}}
$$

When a gas gets cold enough, each atom's wave grows so long that neighboring atoms overlap, and quantum rules take over.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/thermal-de-broglie-wavelength.md

## [Energy-Momentum Relation](https://formulabro.yunusemrevurgun.com/quantum-mechanics/relativistic-energy-momentum/)

$$
E^2 = (p c)^2 + (m c^2)^2
$$

Energy, momentum and mass are linked by one equation, and a particle with no mass always carries energy only through its momentum.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/relativistic-energy-momentum.md

## [Pair Production Threshold](https://formulabro.yunusemrevurgun.com/quantum-mechanics/pair-production-threshold/)

$$
E_\gamma \ge 2 m_e c^2 \approx 1.022 \text{ MeV}
$$

A gamma ray with at least twice the rest energy of an electron can turn into an electron and its antiparticle, a positron.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/pair-production-threshold.md

## [Radioactive Decay Law](https://formulabro.yunusemrevurgun.com/quantum-mechanics/radioactive-decay-law/)

$$
N(t) = N_0\, e^{-\lambda t}
$$

An unstable sample loses a fixed fraction of its atoms in each time interval, so the count falls off in an exponential curve.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/radioactive-decay-law.md

## [Fermi-Dirac Distribution](https://formulabro.yunusemrevurgun.com/quantum-mechanics/fermi-dirac-distribution/)

$$
f(E) = \frac{1}{e^{(E - \mu)/(k_B T)} + 1}
$$

For electrons and other fermions, it gives the chance that an energy level is filled, from 1 for full down to 0 for empty.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/fermi-dirac-distribution.md

## [Bose-Einstein Distribution](https://formulabro.yunusemrevurgun.com/quantum-mechanics/bose-einstein-distribution/)

$$
n(E) = \frac{1}{e^{(E - \mu)/(k_B T)} - 1}
$$

For bosons such as photons, it gives the average number of particles that share a single energy level.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/bose-einstein-distribution.md

## [Fermi Energy](https://formulabro.yunusemrevurgun.com/quantum-mechanics/fermi-energy/)

$$
E_F = \frac{\hbar^2}{2 m}\left(3\pi^2 n\right)^{2/3}
$$

In a metal, electrons fill energy levels from the bottom up, and the Fermi energy is the height of the last level filled.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/fermi-energy.md

## [CHSH Inequality](https://formulabro.yunusemrevurgun.com/quantum-mechanics/chsh-inequality/)

$$
S = E(a,b) - E(a,b') + E(a',b) + E(a',b')
$$

A Bell test combines four measurements into one number, and any theory with pre-set local answers must keep its size at 2 or below.

Markdown: https://formulabro.yunusemrevurgun.com/quantum-mechanics/chsh-inequality.md