Modern Physics

E=mc², photoelectric · 8 formulas

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Mass-Energy Equivalence

E=mc2E = mc^2

Mass is a highly concentrated form of energy; the speed of light squared is the conversion factor.

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Lorentz Factor

γ=11v2/c2\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

As speed approaches c, time stretches and lengths contract by the factor γ.

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Photoelectric Effect

Kmax=hνϕK_{max} = h\nu - \phi

Light comes in quanta of energy hν; only photons above the work-function threshold can free electrons.

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De Broglie Wavelength

λ=hp\lambda = \frac{h}{p}

Every particle has a wavelength inversely proportional to its momentum — heavy/fast things have wavelengths so small they're unobservable.

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Heisenberg Uncertainty Principle

ΔxΔp2\Delta x \cdot \Delta p \geq \frac{\hbar}{2}

Position and momentum cannot both be sharply defined; nature enforces a fundamental fuzziness at small scales.

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Schrödinger Equation (Time-Independent)

22m2ψ+Vψ=Eψ-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi

The total energy operator acting on the wavefunction returns the energy times the same wavefunction — an eigenvalue problem for reality.

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Bohr Energy Levels (Hydrogen)

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

Electrons in hydrogen are stuck on a ladder of negative energies; the gaps determine the colors of light atoms emit.

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Radioactive Decay Law

N(t)=N0eλtN(t) = N_0 e^{-\lambda t}

Each nucleus has a fixed probability per unit time of decaying, producing a smooth exponential decline in the population.