Modern Physics

The factor by which time, length, and relativistic mass change for a moving object.

Moving clocks run slower. Time measured in the rest frame is shorter than time measured by a moving observer.

Moving objects are shortened in the direction of motion as measured by a stationary observer.

Momentum in special relativity, which increases without bound as velocity approaches c.

The most famous equation in physics. Mass and energy are equivalent and interconvertible.

The total energy of a moving particle, including rest energy and kinetic energy.

Kinetic energy in special relativity. Reduces to ½mv² at low velocities.

Relates energy, momentum, and mass. For massless particles (photons): E = pc.

How velocities add in special relativity. Ensures no velocity exceeds c.

The relativistic formula for frequency shift due to relative motion. For approaching sources (v negative), frequency increases (blueshift). For receding sources (v positive), frequency decreases (redshift).

Proper time is the time measured by a clock moving with the object. It is always shorter than coordinate time measured by a stationary observer.

Events that are simultaneous in one frame may not be simultaneous in another. Two events separated in space will have different time separations in different reference frames.

Energy of a photon is proportional to its frequency. Foundation of quantum theory.

Maximum kinetic energy of ejected electrons equals photon energy minus the work function.

The minimum potential needed to stop the most energetic photoelectrons.

All matter has wave-like properties. The wavelength is inversely proportional to momentum.

Position and momentum cannot both be precisely known. Fundamental limit of measurement.

Energy and time cannot both be precisely determined. Related to spectral line widths.

The fundamental equation of quantum mechanics describing how quantum states evolve in time.

Quantized energy levels for a particle confined to a one-dimensional infinite potential well.

Wavelength shift of photons scattered by electrons. Confirms particle nature of light.

Photons carry momentum despite having no rest mass. This is key to understanding radiation pressure and Compton scattering.

The highest occupied energy level at absolute zero for a system of fermions. Crucial for understanding metals, white dwarfs, and neutron stars.

The minimum frequency of light required to eject electrons from a material. Below this frequency, no photoelectric effect occurs regardless of intensity.

Condition for constructive interference in X-ray diffraction from crystal planes. Used to determine crystal structures and to create X-ray spectrometers.

Wavelength of an electron accelerated through potential V. Used in electron microscopy and diffraction experiments.

The most probable distance between electron and nucleus in a hydrogen atom ground state.

Energy levels of the hydrogen atom. Negative values indicate bound states.

Predicts wavelengths of hydrogen spectral lines. n₁ = 1: Lyman, n₁ = 2: Balmer series.

Generalized Bohr energy formula for hydrogen-like atoms (single electron). Energy scales with Z² making inner electrons in heavy atoms much more tightly bound.

The radius of electron orbits in hydrogen-like atoms. Larger n means larger orbits; larger Z means smaller orbits due to stronger nuclear attraction.

Relates characteristic X-ray frequencies to atomic number. The screening constant b accounts for shielding by inner electrons (b ≈ 1 for Kα lines, b ≈ 7.4 for Lα lines).

The number of radioactive nuclei decreases exponentially with time.

The time required for half of the radioactive nuclei to decay.

The rate of radioactive decay, measured in becquerels (decays per second).

Energy required to disassemble a nucleus into separate nucleons.

The energy released in alpha decay equals the mass difference times c². A positive Q-value means the decay is energetically allowed.

Beta-minus decay: a neutron converts to a proton, emitting an electron and antineutrino. This increases the atomic number by 1 while keeping mass number constant.

Nuclear radius scales as the cube root of mass number, implying constant nuclear density. R₀ ≈ 1.2 × 10⁻¹⁵ m (1.2 femtometers).

Total power radiated by an ideal blackbody (emissivity ε = 1) is proportional to the fourth power of temperature. For real (gray) surfaces, use the general emissivity form P = εσAT⁴.

The wavelength at which a blackbody emits most strongly is inversely proportional to temperature.