Quantum

Wave-particle duality · 8 formulas

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de Broglie Wavelength

λ=hp\lambda = \frac{h}{p}

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

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Photon Energy (Planck Relation)

E=hfE = hf

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

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Photoelectric Effect

Kmax=hf−ϕK_{max} = hf - \phi

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

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Heisenberg Uncertainty Principle

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

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

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Bohr Hydrogen Energy Levels

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

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

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Time-Dependent Schrödinger Equation

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

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

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Particle in a 1D Box

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

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

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Dirac Equation

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

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