Quantum and Atomic Physics Cheat Sheet

The core ideas of Quantum and Atomic Physics distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Photoelectric Effect

The emission of electrons from a metal surface when light of sufficiently high frequency shines on it. Einstein explained this by proposing that light consists of photons with energy $E = hf$. Electrons are ejected only if $hf \geq \phi$ (the work function). The maximum kinetic energy of ejected electrons is $KE_{\max} = hf - \phi$.

Photon Energy

Light is quantized into particles called photons, each carrying energy $E = hf = hc/\lambda$, where $h = 6.626 \times 10^{-34}$ J s is Planck's constant, $f$ is frequency, and $\lambda$ is wavelength. Photons also carry momentum $p = h/\lambda = E/c$.

Wave-Particle Duality

All quantum entities exhibit both wave-like and particle-like behavior depending on the experimental context. Light shows wave behavior (interference, diffraction) and particle behavior (photoelectric effect, Compton scattering). Electrons show particle behavior (tracks in detectors) and wave behavior (electron diffraction patterns).

De Broglie Wavelength

Every particle with momentum $p$ has an associated wavelength $\lambda = h/p = h/(mv)$, where $m$ is mass and $v$ is velocity. This wavelength becomes significant only for very small particles (electrons, neutrons) because Planck's constant $h$ is extremely small.

Bohr Model of the Hydrogen Atom

A model in which the electron orbits the proton in quantized circular orbits with angular momentum $L = n\hbar$, where $n = 1, 2, 3, \ldots$ is the principal quantum number. The energy levels are $E_n = -13.6/n^2$ eV. Transitions between levels emit or absorb photons with energy $E = |E_i - E_f|$.

Emission and Absorption Spectra

An emission spectrum consists of bright lines at specific wavelengths produced when atoms emit photons during transitions from higher to lower energy levels. An absorption spectrum shows dark lines at the same wavelengths where atoms absorb photons from a continuous spectrum, exciting electrons to higher levels. Each element has a unique spectral fingerprint.

Heisenberg Uncertainty Principle

A fundamental quantum limit stating that certain pairs of physical properties cannot both be known simultaneously with arbitrary precision: $\Delta x \cdot \Delta p \geq \hbar/2$. This is not a measurement limitation but an intrinsic property of nature. A similar relation holds for energy and time: $\Delta E \cdot \Delta t \geq \hbar/2$.

Radioactive Decay

The spontaneous transformation of an unstable nucleus into a more stable configuration by emitting particles or radiation. Alpha decay emits $^4_2$He nuclei, beta decay converts a neutron to a proton (or vice versa) with emission of an electron (or positron) and neutrino, and gamma decay emits high-energy photons. Each type follows conservation of charge, mass number, and energy.

Half-Life

The time required for half of a radioactive sample to decay: $N(t) = N_0 \cdot (1/2)^{t/t_{1/2}} = N_0 e^{-\lambda t}$, where $\lambda = \ln 2 / t_{1/2}$ is the decay constant. Half-life is a statistical property of large numbers of nuclei; individual decay events are random and unpredictable.

Key Terms at a Glance

Photon:A quantum of electromagnetic radiation (light). A massless particle with energy $E = hf$ and momentum $p = h/\lambda$.

Photoelectric Effect:The ejection of electrons from a surface by light above a threshold frequency. Explained by Einstein's photon model: $KE_{\max} = hf - \phi$.

Work Function:The minimum energy ($\phi$) needed to free an electron from a metal surface. Depends on the material.

Planck's Constant:A fundamental constant: $h = 6.626 \times 10^{-34}$ J s. Relates energy to frequency ($E = hf$) and momentum to wavelength ($p = h/\lambda$).

De Broglie Wavelength:The wavelength associated with a moving particle: $\lambda = h/p = h/(mv)$. Significant only for particles with very small mass.

Wave-Particle Duality:The quantum principle that all entities exhibit both wave and particle properties depending on the experimental context.

Bohr Model:A model of the hydrogen atom with quantized circular orbits and discrete energy levels: $E_n = -13.6/n^2$ eV.

Energy Level:A discrete, quantized energy that an electron can have in an atom. Transitions between levels produce or absorb photons.

Emission Spectrum:Discrete bright lines produced when atoms emit photons during downward electron transitions. Each element has a unique emission spectrum.

Absorption Spectrum:Dark lines in a continuous spectrum where atoms have absorbed photons, exciting electrons to higher energy levels.

Heisenberg Uncertainty Principle:A fundamental quantum limit: $\Delta x \cdot \Delta p \geq \hbar/2$. Position and momentum cannot both be known precisely at the same time.

Alpha Decay:Radioactive decay in which a nucleus emits a helium-4 particle ($^4_2$He), decreasing $A$ by 4 and $Z$ by 2.

Beta Decay:Radioactive decay involving the conversion of a neutron to a proton ($\beta^-$) or proton to neutron ($\beta^+$), with emission of an electron/positron and neutrino.

Gamma Decay:Emission of a high-energy photon from an excited nucleus. Does not change the atomic number or mass number.

Half-Life:The time required for half of a radioactive sample to decay: $t_{1/2} = \ln 2 / \lambda$. A statistical property of large ensembles.

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