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24 formulas

Quantum

Wave-particle duality. Every formula below opens into a live, hands-on simulation.

wave particle duality
λ=hp\lambda = \frac{h}{p}

de Broglie Wavelength

Momentum and wavelength are inversely related through Planck's constant — big things have unmeasurably tiny wavelengths.

quantization
E=hfE = hf

Photon Energy (Planck Relation)

Light energy is quantized: each photon carries a fixed packet of energy set by its frequency.

quantization
Kmax=hf−ϕK_{max} = hf - \phi

Photoelectric Effect

A photon gives all its energy to one electron; the work function is the minimum escape cost.

foundations
Δx Δp≥ℏ2\Delta x \, \Delta p \geq \frac{\hbar}{2}

Heisenberg Uncertainty Principle

Position and momentum are conjugate — pinning one down spreads the other.

atomic physics
En=−13.6 eVn2E_n = -\frac{13.6 \text{ eV}}{n^2}

Bohr Hydrogen Energy Levels

Bound electrons can only sit on a quantized energy ladder; jumping down emits a photon.

foundations
iℏ∂Ψ∂t=H^Ψi\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi

Time-Dependent Schrödinger Equation

The Hamiltonian generates time evolution of the wavefunction in complex Hilbert space.

bound states
En=n2π2ℏ22mL2E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}

Particle in a 1D Box

Standing waves must fit inside the box; only integer half-wavelengths are allowed.

relativistic qm
(iγμ∂μ−mc/ℏ)ψ=0(i\gamma^\mu \partial_\mu - mc/\hbar)\psi = 0

Dirac Equation

A first-order relativistic wave equation whose solutions naturally carry spin and antiparticles.

wave mechanics
H^ψ=Eψ\hat{H}\psi = E\psi

Time-Independent Schrödinger Equation

A wavefunction is allowed only if applying the Hamiltonian gives back the same wavefunction scaled by a number — that number is the energy.

wave mechanics
En=ℏω(n+12)E_n = \hbar\omega\left(n + \tfrac{1}{2}\right)

Quantum Harmonic Oscillator

Energy comes in equal steps of hbar*omega, with a built-in floor of hbar*omega/2 — the zero-point motion required by Heisenberg.

atomic spectra
1λ=RH(1n12−1n22)\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Rydberg Formula

Light emerges when an electron drops between two energy rungs — the wavelength is set by the difference of two inverse squares.

many body
Ψ(r1,r2)=−Ψ(r2,r1)\Psi(\mathbf{r}_1,\mathbf{r}_2) = -\Psi(\mathbf{r}_2,\mathbf{r}_1)

Pauli Exclusion Principle

Swap two identical fermions and the wavefunction flips sign — so the wavefunction vanishes if they share the same state.

spin
S^i=ℏ2σi\hat{S}_i = \tfrac{\hbar}{2}\sigma_i

Spin-1/2 Operators

Three 2×2 matrices encode every spin-1/2 measurement — they don't commute, which is why spin in x and y can't be known simultaneously.

wave mechanics
T≈exp⁡ ⁣(−2ℏ∫ab2m(V(x)−E) dx)T \approx \exp\!\left(-\frac{2}{\hbar}\int_a^b\sqrt{2m\bigl(V(x)-E\bigr)}\,dx\right)

Quantum Tunneling Probability

The wavefunction decays exponentially inside a barrier — make the barrier thinner or shorter and a measurable tail emerges on the far side.

spin
Fz=μz ∂Bz∂zF_z = \mu_z\,\frac{\partial B_z}{\partial z}

Stern–Gerlach Deflection

A magnetic dipole in a field gradient feels a force along the gradient; quantum spin offers only two values of mu_z, so the beam splits in two.

open systems
ρ^=∑ipi ∣ψi⟩⟨ψi∣\hat{\rho} = \sum_i p_i\,|\psi_i\rangle\langle\psi_i|

Density Matrix

Replace one wavefunction with a weighted bookkeeper of many — diagonal entries are probabilities, off-diagonal entries are coherences that decoherence destroys.

foundations
P(x) dx=∣ψ(x)∣2 dxP(x)\,dx = |\psi(x)|^2\,dx

Born Rule

The squared magnitude of the wavefunction is the probability map for measurement outcomes.

operator algebra
[x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar

Canonical Commutation Relation

Order matters: measuring x then p differs from p then x by exactly iℏ — the seed of uncertainty.

angular momentum
∣L∣=l(l+1) ℏ,Lz=mℏ|\mathbf{L}| = \sqrt{l(l+1)}\,\hbar, \qquad L_z = m\hbar

Quantization of Angular Momentum

Angular momentum comes in rungs of ℏ: its length is √(l(l+1))ℏ and its z-shadow is mℏ.

spin dynamics
ωL=γB\omega_L = \gamma B

Larmor Precession

A spin in a magnetic field precesses like a gyroscope, at a rate exactly proportional to the field.

dynamics
d⟨x^⟩dt=⟨p^⟩m,d⟨p^⟩dt=−⟨∂V∂x⟩\frac{d\langle \hat{x}\rangle}{dt} = \frac{\langle \hat{p}\rangle}{m}, \qquad \frac{d\langle \hat{p}\rangle}{dt} = -\left\langle \frac{\partial V}{\partial x}\right\rangle

Ehrenfest Theorem

Quantum averages obey Newton-like equations — the wavepacket's center moves classically.

two level systems
Pe(t)=Ω2Ω2+Δ2 sin⁡2 ⁣(12Ω2+Δ2  t)P_e(t) = \frac{\Omega^2}{\Omega^2 + \Delta^2}\,\sin^2\!\left(\tfrac{1}{2}\sqrt{\Omega^2+\Delta^2}\;t\right)

Rabi Oscillations

A driven two-level system cycles between ground and excited state; detuning caps the swing.

entanglement
S=E(a,b)−E(a,b′)+E(a′,b)+E(a′,b′),∣S∣≤2S = E(a,b) - E(a,b') + E(a',b) + E(a',b'), \qquad |S| \le 2

Bell–CHSH Inequality

Local hidden variables cap a four-correlation sum at 2; entangled particles push it to 2√2.

particle physics
Lint=−e ψˉγμψAμ\mathcal{L}_{\text{int}} = -e\,\bar{\psi}\gamma^{\mu}\psi A_{\mu}

Standard Model Interactions

Every particle reaction in nature is built from a handful of vertices — an electron emitting a photon, a quark emitting a W boson, a gluon splitting. Glue these elementary moves together and you get beta decay, annihilation, Compton scattering, pair production. The bookkeeping rules are absolute: electric charge, lepton number, and baryon number in must equal out. If a reaction conserves them all, somewhere in the universe it happens.