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A neutral hydrogen atom contains one proton and one bound electron. Its simple two-body structure makes it a central test of quantum mechanics. In the nonrelativistic Coulomb potential model, separation of center-of-mass motion leaves a relative coordinate with reduced mass $\mu =m_e{m}_p/(m_e+m_p)$. Its bound energy levels areTheir differences determine emitted or absorbed photon frequencies through $h\nu =|E_n-E_{n^{\prime }}|$. The Bohr radius gives the corresponding characteristic length scale; an energy eigenstate is a spatial wave function, rather than a classical point tracing an orbit.

Resolved spectral lines also reveal the limits of this model. Fine structure, hyperfine structure and the Lamb shift separate levels that a purely nonrelativistic Coulomb calculation leaves degenerate. The Lamb shift experiment uses a microwave-induced change of atomic state and a state-sensitive detector, showing how a small spectroscopic difference can test a much more detailed theory than the leading energy formula.

A uniformly charged spherical shell produces zero field inside and the field of a point charge outside.

An ideal closed loop with negligible resistance can sustain a current that compensates changes in externally linked magnetic flux. In a fixed-geometry circuit model with constant self-inductance $L$, Faraday's law of induction gives $L\dot{I}+{\dot{\Phi }}_{\mathrm{ext}}=0$, hence ${\Phi }_{\mathrm{ext}}+LI$ remains constant.

The conserved flux includes the loop's self-field. Increasing external flux need not increase the total linked flux: the induced current supplies the compensating contribution. With finite resistance, the circuit equation instead contains $RI$, permitting decay and resistive heating.

For an actual superconductor, quantum phase constraints introduce fluxoid and flux quantization in a superconductor. Significant kinetic inductance requires keeping the carrier-motion contribution to the fluxoid; a phase slip can change the winding sector. The simple constant-flux model is therefore a useful idealization with stated circuit assumptions.^[ref]

Electromagnetic damping removes mechanical energy through induced electric currents and their associated magnetic forces. Motion changes the linked magnetic flux or moves a conductor through a field; the induced current then couples the motion to resistive heating.

A conducting rod of length $\ell$ sliding on rails through a perpendicular uniform magnetic field $B$ provides a simple model. With total circuit resistance $R$ and negligible inductance, $\mathcal{E}=B\ell v$, $I=B\ell v/R$ and $F=-B^2{\ell }^{2}v/R$. The mechanical power loss is $B^2{\ell }^2{v}^2/R=I^{2}R$, accounting for where the lost energy goes.

The instantaneous-current model requires the electrical relaxation time to be short compared with the motion's variation time. Significant inductance stores magnetic energy and changes the transient response. Replacing the resistor with an ideal lossless circuit therefore changes the energy balance rather than preserving the same damping law.

Electromagnetic boundary conditions constrain the electric field and magnetic field at the boundary of a region. For a material interface, the interface conditions for electromagnetic fields relate the fields on its two sides. With unit normal $\hat{\mathbf{n}}$ directed from medium 1 to medium 2, free surface charge density ${\sigma }_f$ and free surface current density ${\mathbf{K}}_f$, the usual macroscopic Maxwell equations implyHere $\mathbf{D}$ is the electric displacement field, $\mathbf{B}$ the magnetic flux density, and $\mathbf{H}$ the magnetic field intensity. The relations follow by shrinking a pillbox or rectangular loop across the interface, assuming finite bulk fields and no magnetic surface sources. Material relations such as $\mathbf{D}=\epsilon \mathbf{E}$ then convert them into conditions on the fields used in a particular problem. Even with no free surface sources, all components of both fields need not be continuous when the material properties change.^[ref]

An effective potential collects coordinate-dependent terms after other degrees of freedom have been eliminated or fixed by conservation laws. It allows the remaining coordinate to be studied as a simpler energy problem. The meaning depends on the reduction; an effective potential in quantum field theory is a different construction from the classical radial example below.

For a constant-mass particle in a central potential $U(r)$ with fixed angular momentum magnitude $L$, the conserved mechanical energy becomesAccessible radii satisfy $E\ge U_{\mathrm{eff}}$; equality gives a turning point when the radial motion reverses. A stationary point supports a circular orbit, and a strict local minimum gives radial stability in this reduced model. For $U=-k/r$ with $k>0$ and $L\ne 0$, differentiation gives $r_0=L^2/(mk)$ and $U_{\mathrm{eff}}^{\prime \prime }(r_0)=k/r_0^3>0$. Thus the centrifugal term can support a stable orbit even though the attractive potential alone has no minimum at positive $r$.

Dimensional analysis checks the physical dimensions of quantities and identifies combinations unchanged by the choice of units. It catches impossible equations and reveals scaling before a full dynamical calculation. Adding two quantities requires the same physical dimension; replacing metres by centimetres must change numerical values consistently while preserving physical predictions.

For the ideal simple pendulum, suppose its oscillation period $T$ depends only on mass $m$, length $\ell$, gravitational acceleration $g$, and dimensionless angular amplitude ${\theta }_0$. Write $T=m^a{\ell }^b{g}^{c}F({\theta }_0)$. Since $[m]=M$, $[\ell ]=L$, and $[g]=L{T}^{-2}$, matching physical dimensions gives $a=0$, $b+c=0$, and $-2c=1$. ConsequentlyThe angular amplitude is dimensionless when expressed in radians. Dimensional analysis establishes the length/gravity scaling and the absence of a mass factor under these assumptions, but cannot determine $F$. The harmonic oscillator equation gives $F\to 2\pi$ at small amplitude; the full simple pendulum supplies its finite-amplitude correction. A model with drag or a finite massive support has additional parameters, so the same dimensional argument does not automatically apply.

The result is useful for comparison: at fixed angular amplitude and $g$, lengths in the ratio $1:4$ give periods in the ratio $1:2$. The ratio $T\sqrt{g/\ell }$ should be the same for both in this ideal model. A failure to collapse onto the same value points to measurement error or an omitted parameter. Correct physical dimensions are a necessary consistency check, while the coefficient and the adequacy of the model require more information.

Conservation of energy states that the total energy of an isolated system does not change. For a system that exchanges energy with its surroundings, the corresponding statement is a balance between stored energy and the transfers across the chosen boundary. The relevant system boundary and included forms of energy must therefore be specified.

For a constant-mass particle under a time-independent conservative force $\mathbf{F}=-\nabla U$, Newton's second law givesIts mechanical energy is conserved. With friction, that mechanical quantity generally decreases while internal energy or transfers to the surroundings account for the difference. A damped swinging body becoming stationary is therefore not a failure of conservation of energy. Richard Feynman introduces the principle through accounting for a quantity that can be hidden in different physical forms.^[ref]

The Compton wavelength of a subatomic particle of rest mass $m>0$ is ${\lambda }_C=h/(mc)$, where $h$ is the Planck constant and $c$ the speed of light. It is the wavelength of a photon whose energy $hc/{\lambda }_C$ equals that subatomic particle's rest energy $m{c}^2$. It is a characteristic scale, rather than a measured geometric radius of the subatomic particle.

The reduced Compton wavelength is ${\bar{\lambda }}_C=\hslash /(mc)={\lambda }_C/(2\pi )$, with $\hslash =h/(2\pi )$. Keep that factor of $2\pi$: the Compton scattering wavelength shift below uses $h/(m_{e}c)$, not $\hslash /(m_{e}c)$. Here we choose the rounded electron value ${\lambda }_C=2.426\mathrm{pm}$, where $1\mathrm{pm}=10^{-12}\mathrm{m}$; the maximum single-collision wavelength increase is $2{\lambda }_C=4.852\mathrm{pm}$.

Why this scale? A photon of wavelength $\lambda$ has momentum magnitude $h/\lambda$. At $\lambda \sim {\lambda }_C$ that momentum is comparable with $mc$, so transfer of an appreciable fraction of it can make the recoiling subatomic particle relativistic. Even for longer wavelengths, Compton scattering can resolve the small energy transferred into electron kinetic energy.

A central force on a particle points along the line from a fixed centre to the particle, with magnitude depending only on its distance $r$: $\mathbf{F}=f(r)\hat{\mathbf{r}}$. Its torque about that centre vanishes, since $\mathbf{r}\times \mathbf{F}=0$, so angular momentum is conserved. For nonzero angular momentum, the position vector remains in the plane perpendicular to that fixed vector.

A central potential $U(r)$ gives $f(r)=-dU/dr$. Attractive Newtonian gravity has $U=-GMm/r$; the same inverse-distance form describes the attraction of opposite electric charges through a Coulomb potential. Their radial effective potential separates the physical attraction from the contribution of fixed angular momentum, making turning points and circular orbits accessible without solving every coordinate at once.^[ref]

Added mass is the effective inertia of fluid set in motion when an immersed body accelerates. The body's changing flow field requires kinetic energy; integrating the associated pressure over its surface gives an additional acceleration-dependent force, even in an inviscid model.

For a sphere of volume $V$ accelerating in otherwise stationary, unbounded, incompressible ideal fluid of mass density $\rho$, the added mass is $m_a=\rho V/2$. The fluid's acceleration-dependent force is $-m_a\dot{U}$, so a horizontal force balance takes the form $(m+m_a)\dot{U}=F$ when other forces are absent.^[ref]

An oscillating-body experiment separates this inertial response from velocity-dependent drag by comparing force with both acceleration and velocity. Walls and free surfaces change the flow, and a nonspherical body can have direction-dependent added mass described by a tensor; displaced-fluid mass alone is therefore not a universal coefficient.

For coprime $m,n$, reduction gives $(\mathbb{Z}/mn\mathbb{Z}{)}^{\times }\cong (\mathbb{Z}/m\mathbb{Z}{)}^{\times }\times (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$.

A nonzero polynomial of degree $d$ over a field has at most $d$ roots. In particular, if a polynomial congruence $f(x)\equiv 0(\mathrm{mod}p)$ has degree $d$ and not all coefficients are divisible by the prime $p$, it has at most $d$ incongruent solutions modulo $p$.

Supersymmetry extends spacetime symmetry by odd supercharges that interchange boson and fermion states. Their Super-Poincaré algebra pairs positive-energy states into supermultiplets and gives energy positivity in global supersymmetry. Superspace and superfields package the component fields into representations of this symmetry.

String theory replaces fundamental point-like objects by one-dimensional strings whose motion sweeps out a two-dimensional worldsheet. Quantizing the string's vibration modes produces different particle states; the string tension fixes the energy cost of stretching the string and the associated excitation scale.

A closed-string spectrum contains a massless spin-two mode identified with a graviton, motivating the use of string theory in quantum gravity. Consistent models also involve additional fields, dimensions and symmetry constraints. D-branes supply dynamical objects on which open-string endpoints can lie, and string dualities relate descriptions with different geometrical or coupling interpretations.

These are mathematical and physical structures of the theory, rather than evidence that strings have been directly resolved experimentally. Connecting a specific model to observed low-energy physics requires additional choices of background and compactification, with predictions tested against measurements. The bosonic string theory gives a useful introductory model but lacks several ingredients needed for a realistic particle spectrum.

Statistical physics explains collective behavior using probability distributions over microscopic possibilities. Its central surprise is that incomplete microscopic knowledge can still produce precise predictions for internal energy, entropy and heat capacity. The bridge is statistical mechanics: specify the available microstates, assign their probabilities, then compute the expected values of measured quantities.

A two-level system already displays the idea. Warming it makes its higher energy more likely; its heat capacity first rises, then falls as the two probabilities approach equality. With two distinguishable units, merely counting microstates explains why the middle energy has twice the state degeneracy. These small examples expose the machinery before the much larger sums needed for a photon gas or an Ising model.

The appeal is economical: the same microscopic energies and probabilities explain both an average reading and its variance. A measurement's spread therefore becomes useful physical information rather than an inconvenience to discard.

Special relativity describes physics in Minkowski spacetime, where inertial frames assign different space and time coordinates to the same spacetime events. Its two postulates are Einstein's principle of relativity and the same vacuum speed of light $c$ in every inertial frame, independent of the source's motion. Together they replace the Galilean transformation with the Lorentz transformation, coupling space and time rather than providing a universal clock. Einstein's 1905 paper developed this account through measuring rods, clocks, and light signals. ^[ref].

A spacetime event is an occurrence at a definite place and time: a pulse reaching a detector, for example. An inertial frame supplies a Cartesian coordinate system and clocks at rest throughout it. Einstein synchronisation sets a distant clock's reflection reading to the midpoint between a light signal's departure and return readings at the sending clock. The resulting coordinate time records the local synchronized clock at each spacetime event; it differs from the time when a distant observer receives its image. This distinction makes time dilation, length contraction, and the relativity of simultaneity precise measurements. ^[ref].

The beauty of special relativity is that these different measurements fit one geometry: all inertial frames agree on the Minkowski interval. It lets us translate an experiment between a laboratory and a moving apparatus while preserving the light-signal relations between spacetime events.

Quantum theory supplies a common language for quantum mechanics, measurement in quantum mechanics, and quantum information: prepare a quantum state, transform it, and ask which experimental outcomes occur. In a finite-dimensional idealization, a pure state is a normalized vector of complex numbers. Its entries are probability amplitudes; the Born rule turns their squared magnitudes into probabilities. Between measurements, an isolated system evolves by a unitary operator, preserving its norm and inner products. A quantum circuit makes this evolution explicit as a sequence of quantum gates.

The extra information carried by quantum phases is essential. Relative quantum phases affect how probability amplitudes add when different contributions reach the same output. Their interference can enhance one outcome and cancel another. A global phase, multiplying the entire quantum state by the same complex factor of magnitude one, changes neither these relative phases nor observable predictions. The Hadamard gate below provides a small experiment in which this distinction becomes a deterministic change of output.

Quantum coherence describes this capacity to interfere relative to a specified orthonormal basis. In a density matrix, it appears in off-diagonal entries; a diagonal mixed state retains the populations but lacks that basis coherence. Thus matching the probabilities of one measurement need not match the predictions of a later experiment. Baumgratz, Cramer, and Plenio made this basis-dependent viewpoint precise in their resource framework for quantum coherence. ^[ref].

This makes quantum information an operational question: which distinctions can a preparation, an available unitary operator, and measurement in quantum mechanics reveal? Amplitude amplification illustrates how carefully arranged interference converts a weakly represented answer into a likely observable result. Preserving the relevant quantum coherence throughout the computation is part of the physical task. ^[ref].

Quantum mechanics describes a quantum system using a quantum state, with observables represented by self-adjoint operators. For a single particle in one dimension, a pure state can be described by a wavefunction $\psi (x,t)$: its squared modulus gives the probability density for position. The Schrödinger equation predicts how this probability amplitude evolves between interventions; measurement in quantum mechanics connects it to experimental outcomes through the Born rule. A wavefunction therefore predicts a distribution of possible positions rather than a single trajectory as in classical mechanics.

The infinite square well makes three central ideas visible with elementary sine functions. Confinement selects discrete energy eigenvalues; wavefunction normalization converts a shape into quantitative probabilities; and a quantum superposition can have a changing probability density even when each constituent energy eigenstate has a stationary one. This combination explains why quantum mechanics uses both linear algebra and differential equations: eigenfunctions describe the building blocks, while quantum phases determine how they interfere. The same tools help explain the discrete energies of chemical atoms and molecules, although their Quantum Hamiltonians require different potentials and often many particles.

In his first 1926 paper, Erwin Schrödinger formulated quantization as an eigenvalue problem for a hydrogen atom in nonrelativistic quantum mechanics. On its opening page he compared the emergence of integers with the number of nodes of a standing wave on a vibrating string. The infinite square well below realizes that appealing idea especially transparently: the allowed integers follow from the boundary conditions. This historical analogy concerns the origin of discrete energies; the Born rule supplies the statistical interpretation of the modern wavefunction. ^[ref].

Quantum field theory describes quantum states and interactions using fields whose smeared values act as operators. Free-field modes behave as quantum harmonic oscillators or fermionic occupation modes; their excitations supply the particle description. This allows creation operators and annihilation operators to describe processes in which particle number changes.

A relativistic quantum field theory combines this framework with special relativity, Lorentz invariance and locality conditions such as microcausality. Nonrelativistic many-body systems also have useful quantum field theories, for example a weakly interacting Bose gas described by a bosonic field. The framework is therefore broader than a list of elementary-particle models.

Correlation functions predict responses and connect through appropriate limits to scattering amplitudes. Perturbative quantum field theory expands these quantities using Feynman diagrams, while strongly interacting problems may require other methods. Composite fields and arbitrarily short distances require attention to operator definitions and renormalization, rather than treating every field product as an ordinary pointwise multiplication.