--- title: "Quantum & Modern Physics" description: "42 topics in Science, in the order they build on each other." canonical: https://lightmysky.com/learn/science/areas/quantum-and-modern-physics source: https://lightmysky.com/learn/science/areas/quantum-and-modern-physics.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Quantum & Modern Physics 42 topics in Science, in the order they build on each other. Page: https://lightmysky.com/learn/science/areas/quantum-and-modern-physics - [Blackbody Radiation and the Quantum of Energy](https://lightmysky.com/learn/science/blackbody-radiation-and-the-quantum-of-energy-mt_B2Bk2IAOa9): The measured spectrum of a hot body disagrees with classical physics at short wavelengths, and the disagreement disappears only if energy is exchanged in packets proportional to frequency. Planck's constant enters physics as the size of that packet. - [The Compton Effect and Photon Momentum](https://lightmysky.com/learn/science/the-compton-effect-and-photon-momentum-mt_wt7kdaof_P): X-rays scattered from electrons come back with a longer wavelength that depends only on the scattering angle, which is what a collision between two particles predicts. The photon therefore carries momentum h over lambda. - [The Postulates of Special Relativity and the Loss of Simultaneity](https://lightmysky.com/learn/science/the-postulates-of-special-relativity-and-the-loss-of-simultaneity-mt_kZqVQ18bbc): Taking the laws of physics and the speed of light to be the same for every inertial observer forces two events simultaneous for one observer to be separated in time for another. Nothing else is assumed, and everything else follows. - [The Wavefunction and the Born Probability Rule](https://lightmysky.com/learn/science/the-wavefunction-and-the-born-probability-rule-mt_elzmbBzmKE): A particle is described by a complex wavefunction whose squared magnitude gives the probability of finding it in a small region. Position stops being a property the particle has and becomes something a measurement returns. - [Time Dilation and Length Contraction](https://lightmysky.com/learn/science/time-dilation-and-length-contraction-mt_7tpKiFWeG0): A moving clock runs slow and a moving ruler is short, by the same factor, as measured from the frame the motion is relative to. Muons reaching the ground are the everyday evidence. - [Normalisation and Expectation Values](https://lightmysky.com/learn/science/normalisation-and-expectation-values-mt_6Y3DZ8P1qm): Total probability must be one, which fixes the constant in front of any wavefunction, and averages of position or momentum are integrals weighted by the probability density. Expectation values are what a long run of measurements would average to. - [The Lorentz Transformation and Velocity Addition](https://lightmysky.com/learn/science/the-lorentz-transformation-and-velocity-addition-mt_uG3_gArw19): One set of equations converts coordinates between frames and contains dilation and contraction as special cases. Adding velocities with it never gives a result above the speed of light. - [Relativistic Momentum and Energy](https://lightmysky.com/learn/science/relativistic-momentum-and-energy-mt_zBq87F-j3B): Momentum and energy have to be redefined for conservation to hold in every frame, and the redefinition gives every mass a rest energy. Mass and energy become two readings of the same quantity. - [Spacetime Intervals and Four-Vectors](https://lightmysky.com/learn/science/spacetime-intervals-and-four-vectors-mt_CYdi7Qpvrf): Observers disagree about distances and times but agree on the interval between two events, which is the invariant of spacetime. Grouping quantities into four-vectors makes that invariance automatic. - [Properties of Nuclei and the Nuclear Force](https://lightmysky.com/learn/science/properties-of-nuclei-and-the-nuclear-force-mt_HdMS_muGWa): Nuclear radius grows with the cube root of nucleon number, so nuclear density is nearly constant, and the force holding nucleons together is short ranged and blind to charge. Those two facts shape the binding energy curve. - [The Uncertainty Principle from Wave Packets](https://lightmysky.com/learn/science/the-uncertainty-principle-from-wave-packets-mt_DPYnoGksRd): Localising a particle needs many wavelengths added together, so a narrow position spread forces a wide momentum spread. The limit is a property of waves, not a comment on clumsy apparatus. - [The Time-Independent Schrödinger Equation](https://lightmysky.com/learn/science/the-time-independent-schrodinger-equation-mt_zeBVPery5T): Energy conservation written for a wavefunction gives a differential equation whose solutions are the allowed stationary states. Solving a quantum problem means solving that equation with boundary conditions. - [The Infinite Square Well and Energy Quantisation](https://lightmysky.com/learn/science/the-infinite-square-well-and-energy-quantisation-mt_VV9ItoW7ew): A particle trapped between hard walls can only hold whole numbers of half wavelengths, so its energies form a discrete ladder that rises as the square of the quantum number. Quantisation comes from the boundary conditions, not from an extra rule. - [Decay Modes, Decay Chains and Nuclear Reactions](https://lightmysky.com/learn/science/decay-modes-decay-chains-and-nuclear-reactions-mt_b_imidwJeQ): Which decay a nucleus takes depends on where it sits relative to the line of stability, and a heavy nucleus usually needs a chain of steps to reach it. Reaction equations balance charge, nucleon number and energy. - [Fission Reactors and Fusion in the Laboratory](https://lightmysky.com/learn/science/fission-reactors-and-fusion-in-the-laboratory-mt_iTwgXI0iBy): A chain reaction is controlled by moderating neutrons and absorbing the surplus, while fusion needs temperatures high enough to beat the Coulomb barrier and confinement long enough to pay for itself. The physics of each explains the engineering. - [Nuclear Models: the Liquid Drop and the Semi-Empirical Mass Formula](https://lightmysky.com/learn/science/nuclear-models-the-liquid-drop-and-the-semi-empirical-mass-formula-mt_-cTGNdUJLz): Treating a nucleus as an incompressible charged drop gives volume, surface, Coulomb, asymmetry and pairing terms that reproduce the binding energy curve to within a per cent. The same formula says which nuclide in an isobaric chain is stable and why heavy nuclei fission at all. - [Particle Accelerators, Detectors and Cross-Sections](https://lightmysky.com/learn/science/particle-accelerators-detectors-and-cross-sections-mt_euJtxtxXyV): Finding structure at small scale needs high energy, because resolution is limited by the de Broglie wavelength of the probe. What comes out of a collision is counted as a cross-section rather than watched directly. - [The Finite Well and Wavefunction Penetration](https://lightmysky.com/learn/science/the-finite-well-and-wavefunction-penetration-mt_Vs4pWnXW7l): When the walls are of finite height the wavefunction does not stop at the edge but decays into the classically forbidden region. The well then holds only a limited number of bound states. - [Quantum Tunnelling and the Transmission Probability](https://lightmysky.com/learn/science/quantum-tunnelling-and-the-transmission-probability-mt_tLdPJKx8Id): A particle meeting a barrier it cannot classically cross has a small chance of appearing on the far side, falling off exponentially with barrier width and height. Alpha decay and the scanning tunnelling microscope both live on that exponential. - [The Quantum Harmonic Oscillator and Zero-Point Energy](https://lightmysky.com/learn/science/the-quantum-harmonic-oscillator-and-zero-point-energy-mt_HyDxmqyReX): A parabolic potential gives evenly spaced energy levels, and the lowest of them sits above the bottom of the well. A quantum oscillator can never be brought completely to rest. - [Operators, Eigenvalues and Measurement](https://lightmysky.com/learn/science/operators-eigenvalues-and-measurement-mt_E9C_9DlwD2): Every observable is represented by an operator, and the values a measurement can return are that operator's eigenvalues. A state that is not an eigenstate gives one of those values with a probability set by the wavefunction. - [Separating the Schrödinger Equation for Hydrogen](https://lightmysky.com/learn/science/separating-the-schrodinger-equation-for-hydrogen-mt_Yk2qHB0_v2): In a spherically symmetric potential the wavefunction splits into a radial part and an angular part, and each piece brings its own quantum number. The energy levels of hydrogen come out of the radial equation alone. - [The Standard Model and the Four Interactions](https://lightmysky.com/learn/science/the-standard-model-and-the-four-interactions-mt_GYB4UMS0D9): Matter is built from quarks and leptons in three generations, and forces are carried by exchange particles with characteristic ranges and strengths. The model states which interactions each particle feels. - [Orbital Angular Momentum and the Quantum Numbers](https://lightmysky.com/learn/science/orbital-angular-momentum-and-the-quantum-numbers-mt_OXlbat0vat): Angular momentum is quantised in both size and direction, so only certain projections along a chosen axis are allowed. The shapes labelled s, p and d are the angular solutions drawn out. - [Electron Spin and the Stern-Gerlach Experiment](https://lightmysky.com/learn/science/electron-spin-and-the-stern-gerlach-experiment-mt_HVFLoNe6wk): A beam of silver atoms splits into exactly two in a non-uniform magnetic field, which no orbital motion can explain. Electrons carry an intrinsic angular momentum with only two possible projections. - [The Exclusion Principle and Multi-Electron Atoms](https://lightmysky.com/learn/science/the-exclusion-principle-and-multi-electron-atoms-mt_4tLa-3Av0A): No two electrons in an atom share all four quantum numbers, so electrons stack into shells instead of collapsing into the lowest state. The periodic table is that stacking rule made visible. - [First-Order Perturbation Theory](https://lightmysky.com/learn/science/first-order-perturbation-theory-mt_UsbiiQFOAt): Most potentials cannot be solved exactly, so a small extra term is treated as a correction to a solved problem. The first-order energy shift is the expectation value of that term in the unperturbed state. - [The Nuclear Shell Model and the Magic Numbers](https://lightmysky.com/learn/science/the-nuclear-shell-model-and-the-magic-numbers-mt_1tiXMm0sA2): The drop model says nothing about why nuclei with 2, 8, 20, 28, 50 or 82 of either nucleon are unusually tightly bound. Putting nucleons in levels of a mean potential, with a strong spin-orbit term, produces exactly those gaps and predicts ground-state spins. - [States as Vectors: Dirac Notation and the State Space](https://lightmysky.com/learn/science/states-as-vectors-dirac-notation-and-the-state-space-mt_9g0aNMxQ8M): A quantum state is a vector in a complex space carrying an inner product, and the wavefunction is only its components in one particular basis. Bra-ket notation writes the same state in position, momentum or energy language without changing the state itself. - [Observables as Hermitian Operators and Their Spectra](https://lightmysky.com/learn/science/observables-as-hermitian-operators-and-their-spectra-mt_TXCN8uiWqs): Every measurable quantity is a Hermitian operator whose eigenvalues are the possible results and whose eigenvectors span the state space. Continuous spectra need a different normalisation, which is why a momentum eigenstate is not an ordinary state. - [Commutators, Compatible Observables and the General Uncertainty Relation](https://lightmysky.com/learn/science/commutators-compatible-observables-and-the-general-uncertainty-relation-mt_TKoUwu74K9): Two quantities can be known together only when their operators commute, and the commutator sets a floor on the product of their spreads. Position and momentum are one case of a relation that holds for any pair of observables. - [Unitary Time Evolution and the Heisenberg Picture](https://lightmysky.com/learn/science/unitary-time-evolution-and-the-heisenberg-picture-mt_3JsTc9VYj8): Time evolution is a unitary operator generated by the Hamiltonian, which is what keeps total probability fixed. Putting the time dependence on the operators instead of the state gives identical predictions and often less algebra. - [Angular Momentum from Its Commutation Relations](https://lightmysky.com/learn/science/angular-momentum-from-its-commutation-relations-mt_u0y20XJh-v): The entire angular momentum spectrum follows from the commutation relations alone, using raising and lowering operators to step between states. Half-integer values fall out of the algebra, which is where spin belongs. - [Adding Angular Momenta and the Coupled Basis](https://lightmysky.com/learn/science/adding-angular-momenta-and-the-coupled-basis-mt_eSBNPKm9C4): Two angular momenta combine into a total whose allowed values run from the difference to the sum, and the coupled basis is a change of basis from the product one. Atomic fine structure and nuclear spin states are read off this way. - [Identical Particles and the Symmetry of a Many-Body State](https://lightmysky.com/learn/science/identical-particles-and-the-symmetry-of-a-many-body-state-mt_hns-eEGeHU): Swapping two identical particles cannot change any prediction, which leaves only symmetric or antisymmetric states. That single requirement produces the exclusion principle for fermions and the crowding of bosons into one state. - [Degenerate Perturbation Theory and the Variational Method](https://lightmysky.com/learn/science/degenerate-perturbation-theory-and-the-variational-method-mt_uov0PZ8vqy): When unperturbed states share an energy, the correction has to be found by diagonalising the perturbation inside that subspace. The variational method gives a second handle: any trial state puts an upper bound on the ground state energy. - [The Path Integral and the Sum Over Histories](https://lightmysky.com/learn/science/the-path-integral-and-the-sum-over-histories-mt_QRLheflB1O): An amplitude can be written as a sum over every path from start to finish, each contributing a phase equal to its action in units of the Planck constant. Paths near the stationary one add in step and the rest cancel, which is why the classical path appears at all. - [Time-Dependent Perturbation Theory and Fermi's Golden Rule](https://lightmysky.com/learn/science/time-dependent-perturbation-theory-and-fermis-golden-rule-mt_zDSASGF7SC): A perturbation switched on in time drives transitions between states at a rate set by a matrix element and the density of final states. This is where absorption, emission and decay rates come from. - [Scattering: Cross-Sections and the Born Approximation](https://lightmysky.com/learn/science/scattering-cross-sections-and-the-born-approximation-mt_xmN9KDfSp7): A scattering experiment measures a cross-section, which is the area a target seems to present to an incoming beam. Treating the potential as a weak perturbation makes the scattering amplitude a Fourier transform of that potential, so the pattern maps the target. - [Density Matrices, Entanglement and the Bell Test](https://lightmysky.com/learn/science/density-matrices-entanglement-and-the-bell-test-mt_qwUJ2qV0l3): A state that is only known statistically needs a density matrix rather than a state vector, and the two situations it covers, ignorance and entanglement, look identical to anyone holding one half of the system. Bell's inequality turns the difference between quantum correlation and any local explanation into a number an experiment can measure. - [Field Quantisation: Creation Operators and Particle Number](https://lightmysky.com/learn/science/field-quantisation-creation-operators-and-particle-number-mt_BJyZWQA5cb): Treating each mode of a field as a harmonic oscillator makes its quantum number a count of particles, so particles become excitations rather than objects. Creation and annihilation operators then explain why identical particles are identical and why particle number need not be conserved. - [The Klein-Gordon and Dirac Equations](https://lightmysky.com/learn/science/the-klein-gordon-and-dirac-equations-mt_njCW9OEevW): Demanding a wave equation that treats space and time the way relativity does gives a second-order equation with negative-energy solutions and no positive probability density. Insisting on first order in time forces the wavefunction to carry four components, and spin and antiparticles arrive as consequences rather than as additions.