Modern &amp; Quantum Physics

Time Dilation

\Delta t = \frac{\Delta t_0}{\sqrt{1 - v^2/c^2}}

Moving clocks run slow.

Length Contraction

L = L_0\sqrt{1 - v^2/c^2}

Moving objects shorten along motion.

Relativistic Momentum

p = \frac{mv}{\sqrt{1 - v^2/c^2}}

Momentum at speeds near light.

Lorentz Factor

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Scaling factor in special relativity.

de Broglie Wavelength

\lambda = \frac{h}{p}

Wavelength associated with a particle.

Uncertainty Principle

\Delta x\,\Delta p \geq \frac{\hbar}{2}

Limit on knowing position and momentum together.

Planck Relation

E = hf

Quantised energy of radiation.

Photoelectric Effect

K_{max} = hf - \phi

Kinetic energy of ejected electrons.

Bohr Energy Levels

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

Energy of the nth hydrogen orbit.

Radioactive Decay

N = N_0 e^{-\lambda t}

Number of nuclei remaining over time.

Half-Life

t_{1/2} = \frac{\ln 2}{\lambda}

Time for half a sample to decay.

Radioactive Activity

A = \lambda N

Decays per second in a sample.

Binding Energy

E_b = \Delta m\,c^2

Energy holding a nucleus together.

Wien's Displacement Law

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

Peak emission wavelength of a hot body.

Compton Shift

\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Wavelength change of scattered photons.

Schrödinger (Time-Independent)

\hat{H}\psi = E\psi

Fundamental equation of quantum mechanics.