Essential Physics Equations

Explore foundational physics equations, grouped by domain, with explanations, symbol definitions, SI units, constant values, and the conditions in which they apply.

Source code

Classical mechanics

How forces change motion, how energy is transferred, and why momentum is conserved.

Newton’s second law

\sum \mathbf{F} = m\mathbf{a}

The combined force on an object determines its acceleration. A larger mass needs a larger net force to produce the same acceleration. Forces and acceleration have direction, so forces that oppose each other must be combined as vectors. OpenStax · Newton’s second law

Newton’s second law: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\sum \mathbf{F}$	Net external force	$\mathrm{N}$	—
$m$	Mass	$\mathrm{kg}$	—
$\mathbf{a}$	Acceleration	$\mathrm{m\,s^{-2}}$	—

When it applies: This form assumes constant mass and an inertial reference frame. Use it for motion much slower than light.

Conservation of linear momentum

\sum_i \mathbf{p}_{i,\mathrm{initial}} = \sum_i \mathbf{p}_{i,\mathrm{final}}

Objects can exchange momentum during a collision, but their total momentum stays the same when no net external impulse acts on the system. This lets us connect motion before and after a collision without knowing every detail of the impact. OpenStax · Conservation of linear momentum

Conservation of linear momentum: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\mathbf{p}_i$	Momentum of the ith object; classically, its mass multiplied by its velocity	$\mathrm{kg\,m\,s^{-1}}$	—
$\sum_i$	Vector sum over all objects in the chosen system	—	—

When it applies: Choose a system with no mass crossing its boundary and negligible net external impulse during the interval. Kinetic energy need not be conserved.

Kinetic energy

K = \frac{1}{2}mv^2

Kinetic energy is the energy associated with motion. Doubling the speed gives four times the kinetic energy at the same mass. That is why stopping a fast vehicle requires much more energy to be removed than stopping a slow one. OpenStax · Kinetic energy

Kinetic energy: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$K$	Translational kinetic energy	$\mathrm{J}$	—
$m$	Mass	$\mathrm{kg}$	—
$v$	Speed in the chosen reference frame	$\mathrm{m\,s^{-1}}$	—

When it applies: This is the nonrelativistic expression for translational motion. Rotation has its own kinetic energy, and speeds near light require a relativistic expression.

Work–energy theorem

W_{\mathrm{net}} = \Delta K

The total work done on a particle equals its change in kinetic energy. Positive net work speeds it up; negative net work slows it down. Work counts the part of a force acting along the displacement, rather than force alone. OpenStax · Work–energy theorem

Work–energy theorem: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$W_{\mathrm{net}}$	Work done by all forces along the path	$\mathrm{J}$	—
$\Delta K$	Final kinetic energy minus initial kinetic energy	$\mathrm{J}$	—

When it applies: Use the particle model in classical mechanics. For an extended or deforming system, also account for rotation, internal energy, and how the system boundary is chosen.

Newton’s law of gravitation

F = G\frac{m_1m_2}{r^2}

Two masses attract each other with a force that weakens with the square of their separation. The same relationship describes falling objects and much of planetary motion. The force points along the line joining the two masses. OpenStax · Universal gravitation

Newton’s law of gravitation: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$F$	Magnitude of the attractive force	$\mathrm{N}$	—
$G$	Newtonian gravitational constant	$\mathrm{m^3\,kg^{-1}\,s^{-2}}$	$\approx 6.67430\times10^{-11}$
$m_1,\ m_2$	The two masses	$\mathrm{kg}$	—
$r$	Distance between the masses’ centres	$\mathrm{m}$	—

When it applies: Applies directly to point masses or nonoverlapping spherically symmetric bodies. Strong gravity and relativistic motion require general relativity.

Oscillations, waves, and optics

Repeating motion, the propagation of disturbances, and the bending of light.

Simple harmonic motion

\frac{d^2x}{dt^2} = -\omega^2 x

Acceleration always points toward equilibrium and is proportional to the displacement. This produces sinusoidal motion, such as an ideal mass on a spring. The negative sign expresses the restoring direction. OpenStax · Simple harmonic motion

Simple harmonic motion: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$x$	Displacement from equilibrium	$\mathrm{m}$	—
$t$	Time	$\mathrm{s}$	—
$\omega$	Angular frequency	$\mathrm{rad\,s^{-1}}$	—
$d^2x/dt^2$	Second time derivative of displacement: acceleration	$\mathrm{m\,s^{-2}}$	—

When it applies: Assumes a linear restoring force with no damping or driving. Pendulums approximate this motion only for small angles.

Wave speed, frequency, and wavelength

v_{\mathrm{phase}} = f\lambda

A wave crest travels one wavelength during one cycle. Multiplying the number of cycles per second by the distance per cycle gives the crest’s speed. The relationship connects the spatial and temporal patterns of a periodic wave. OpenStax · Mathematics of waves

Wave speed, frequency, and wavelength: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$v_{\mathrm{phase}}$	Phase speed	$\mathrm{m\,s^{-1}}$	—
$f$	Frequency	$\mathrm{Hz}$	—
$\lambda$	Wavelength	$\mathrm{m}$	—

When it applies: Describes a periodic wave. In a dispersive medium, phase speed can differ from the speed of a wave packet or signal.

Snell’s law of refraction

n_1\sin\theta_1 = n_2\sin\theta_2

Light changes direction when it crosses between materials with different refractive indices. Entering a material with a higher index bends the ray toward the surface normal. The law is used to understand lenses and light passing through water or glass. OpenStax · Refraction

Snell’s law of refraction: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$n_1,\ n_2$	Refractive indices of the incident and transmitted media	$1$	—
$\theta_1,\ \theta_2$	Incident and refracted angles, both measured from the surface normal	$\mathrm{rad}$	—

When it applies: Use for refraction at an interface between ordinary isotropic media. Some incident angles from a higher-index medium produce total internal reflection instead of a transmitted ray.

Electromagnetism

Charges, currents, and fields. The four Maxwell equations below connect electricity, magnetism, and light.

Coulomb’s law

F = \frac{1}{4\pi\varepsilon_0}\frac{|q_1q_2|}{r^2}

Electric charges exert forces on each other. Like signs repel and opposite signs attract. The displayed equation gives the force magnitude; its direction lies along the line joining the charges. OpenStax · Coulomb’s law

Coulomb’s law: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$F$	Electric force magnitude	$\mathrm{N}$	—
$q_1,\ q_2$	Electric charges	$\mathrm{C}$	—
$r$	Separation	$\mathrm{m}$	—
$\varepsilon_0$	Vacuum permittivity	$\mathrm{F\,m^{-1}}$	$\approx 8.8541878188\times10^{-12}$
$\pi$	Ratio of a circle’s circumference to its diameter	$1$	$\approx 3.141592653589793$

When it applies: This is the electrostatic force between point charges in vacuum. For many charges, add their force vectors; material media require additional treatment.

Lorentz force

\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)

Electric and magnetic fields act on a charged particle. The electric part can change its speed. The magnetic part is perpendicular to its velocity, so by itself it changes direction without doing work. OpenStax · Maxwell equations and the Lorentz force

Lorentz force: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\mathbf{F}$	Electromagnetic force	$\mathrm{N}$	—
$q$	Particle charge	$\mathrm{C}$	—
$\mathbf{E}$	Electric field at the particle	$\mathrm{V\,m^{-1}}$	—
$\mathbf{B}$	Magnetic field at the particle	$\mathrm{T}$	—
$\mathbf{v}$	Particle velocity; the cross denotes a vector cross product	$\mathrm{m\,s^{-1}}$	—

When it applies: Fields and velocity must be measured in the same inertial frame. Treat the particle as a point charge and exclude its own field from the applied fields.

Gauss’s law for electricity

\oiint_S \mathbf{E}\cdot d\mathbf{A} = \frac{Q_{\mathrm{enc}}}{\varepsilon_0}

The net electric flux through a closed surface measures the charge enclosed by it. Charges outside the surface affect the field but contribute no net flux through that surface. Symmetry can make this a powerful way to find electric fields. OpenStax · Gauss’s law

Gauss’s law for electricity: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\mathbf{E}$	Electric field	$\mathrm{V\,m^{-1}}$	—
$S$	Closed surface of integration	—	—
$d\mathbf{A}$	Outward-pointing area element	$\mathrm{m^2}$	—
$Q_{\mathrm{enc}}$	Net enclosed charge	$\mathrm{C}$	—
$\varepsilon_0$	Vacuum permittivity	$\mathrm{F\,m^{-1}}$	$\approx 8.8541878188\times10^{-12}$

When it applies: One of Maxwell’s equations in SI units, using total charge and the electric field. It does not require a symmetric surface to be valid.

Gauss’s law for magnetism

\oiint_S \mathbf{B}\cdot d\mathbf{A} = 0

A closed surface has no net magnetic flux. Magnetic field lines have no isolated sources or sinks in ordinary electromagnetism. Cutting a bar magnet creates smaller magnets, each with both poles, rather than separating the poles. OpenStax · Magnetic fields and lines

Gauss’s law for magnetism: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\mathbf{B}$	Magnetic field	$\mathrm{T}$	—
$S$	Closed surface of integration	—	—
$d\mathbf{A}$	Outward-pointing area element	$\mathrm{m^2}$	—
$\oiint_S$	Integral summing the normal component of the field over the closed surface	—	—

When it applies: The Maxwell equation for a theory without magnetic monopoles. Zero net flux does not mean the magnetic field vanishes everywhere on the surface.

Faraday’s law of induction

\oint_C \mathbf{E}\cdot d\boldsymbol{\ell} = -\frac{d\Phi_B}{dt}

A changing magnetic flux produces a circulating electric field. This is the basis of transformers and induction. The negative sign records Lenz’s law: the induced response opposes the change in flux. OpenStax · Induced electric fields

Faraday’s law of induction: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\mathbf{E}$	Electric field	$\mathrm{V\,m^{-1}}$	—
$C$	Closed loop of integration	—	—
$d\boldsymbol{\ell}$	Directed line element along the loop	$\mathrm{m}$	—
$\Phi_B$	Magnetic flux through a surface bounded by the loop	$\mathrm{Wb}$	—
$t$	Time; the loop integral is the induced electromotive force	$\mathrm{s}$	—

When it applies: This field form uses a stationary loop with consistent loop and surface orientations. Moving conductors also require the magnetic force contribution to electromotive force.

Ampère–Maxwell law

\oint_C \mathbf{B}\cdot d\boldsymbol{\ell} = \mu_0 I_{\mathrm{enc}} + \mu_0\varepsilon_0\frac{d\Phi_E}{dt}

Magnetic circulation comes from electric current and changing electric flux. Maxwell’s added flux term makes the law work across a charging capacitor and helps explain how electromagnetic waves propagate. OpenStax · Maxwell’s correction to Ampère’s law

Ampère–Maxwell law: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\mathbf{B}$	Magnetic field	$\mathrm{T}$	—
$C$	Closed loop of integration	—	—
$d\boldsymbol{\ell}$	Directed line element along the loop	$\mathrm{m}$	—
$I_{\mathrm{enc}}$	Current through a surface bounded by the loop	$\mathrm{A}$	—
$\Phi_E$	Electric flux through that surface	$\mathrm{V\,m}$	—
$t$	Time	$\mathrm{s}$	—
$\mu_0$	Vacuum permeability	$\mathrm{N\,A^{-2}}$	$\approx 1.25663706127\times10^{-6}$
$\varepsilon_0$	Vacuum permittivity	$\mathrm{F\,m^{-1}}$	$\approx 8.8541878188\times10^{-12}$

When it applies: SI field form for a fixed loop and surface, using total current. Loop direction, surface normal, current, and flux signs follow the right-hand rule.

Ohm’s law

V = IR

Across an ohmic component, the voltage is proportional to the current. Resistance sets how much voltage is needed for a given current, making this a starting point for analysing resistor circuits. OpenStax · Ohm’s law

Ohm’s law: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$V$	Potential difference across the component	$\mathrm{V}$	—
$I$	Current through the component	$\mathrm{A}$	—
$R$	Resistance	$\mathrm{\Omega}$	—

When it applies: Resistance must be effectively constant under the operating conditions. Diodes, lamps with changing temperature, and many other components are not ohmic.

Thermodynamics and statistical physics

Heat, work, temperature, and the connection between microscopic states and macroscopic behaviour.

Ideal gas law

PV = nRT

Pressure, volume, and temperature are linked for a fixed amount of ideal gas. Heating a sealed, rigid container raises its pressure; allowing expansion can instead increase its volume. The relationship is a model of many particles moving and colliding. OpenStax · Molecular model of an ideal gas

Ideal gas law: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$P$	Absolute pressure	$\mathrm{Pa}$	—
$V$	Volume	$\mathrm{m^3}$	—
$n$	Amount of gas	$\mathrm{mol}$	—
$R$	Molar gas constant	$\mathrm{J\,mol^{-1}\,K^{-1}}$	$8.31446261815324$
$T$	Absolute temperature	$\mathrm{K}$	—

When it applies: Assumes negligible particle volume and interactions apart from elastic collisions. Real gases approximate it best at low density and away from condensation.

First law of thermodynamics

\Delta U = Q - W

Energy entering a system as heat can increase its internal energy or leave as work. The first law is an energy balance: heat and work are ways to transfer energy, while internal energy belongs to the system’s state. OpenStax · First law of thermodynamics

First law of thermodynamics: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\Delta U$	Change in internal energy	$\mathrm{J}$	—
$Q$	Heat transferred into the system, positive inward	$\mathrm{J}$	—
$W$	Work done by the system on its surroundings, positive outward	$\mathrm{J}$	—

When it applies: This form assumes a closed system with negligible changes in bulk kinetic and gravitational potential energy. A convention that counts work done on the system reverses the work sign.

Second law of thermodynamics

\Delta S_{\mathrm{isolated}} \geq 0

The total entropy of an isolated system cannot decrease in a macroscopic thermodynamic process. This gives processes a preferred direction: heat spontaneously flows from hotter to colder bodies, and dissipated energy cannot all be recovered as useful work in a cycle. OpenStax · Entropy and the second law

Second law of thermodynamics: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\Delta S_{\mathrm{isolated}}$	Total entropy change of the isolated system	$\mathrm{J\,K^{-1}}$	—
$\geq 0$	Zero for a reversible process; positive for an irreversible process	—	—

When it applies: Include the surroundings if the chosen subsystem exchanges heat or matter. A subsystem’s entropy can decrease while total entropy increases.

Boltzmann’s entropy formula

S = k_{\mathrm{B}}\ln\Omega

A macroscopic state can be realised by many microscopic arrangements. Entropy measures the logarithm of their number. A state compatible with more microscopic arrangements is more likely when those arrangements are equally probable. OpenStax · Entropy on a microscopic scale

Boltzmann’s entropy formula: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$S$	Entropy	$\mathrm{J\,K^{-1}}$	—
$k_{\mathrm{B}}$	Boltzmann constant	$\mathrm{J\,K^{-1}}$	$1.380649\times10^{-23}$
$\Omega$	Number of accessible microstates compatible with the macrostate	$1$	—
$\ln$	Natural logarithm	—	—

When it applies: This counting form assumes equally probable accessible microstates. Unequal probabilities require the more general statistical entropy expression.

Relativity and spacetime

The relationship between mass and energy, the dependence of elapsed time on motion, and gravity as spacetime geometry.

Mass–energy equivalence

E_0 = mc^2

An object has rest energy even when it is not moving. Mass and rest energy are two ways to describe the same physical property. In a nuclear reaction, the difference between initial and final rest masses accounts for energy released; ordinary matter does not readily release all its rest energy. OpenStax · Rest energy and relativistic energy

Mass–energy equivalence: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$E_0$	Rest energy	$\mathrm{J}$	—
$m$	Invariant mass, also called rest mass	$\mathrm{kg}$	—
$c$	Speed of light in vacuum	$\mathrm{m\,s^{-1}}$	$299\,792\,458$

When it applies: Gives rest energy, not the total energy of a moving object. For a composite system, internal motion and binding energy contribute to its invariant mass.

Relativistic energy–momentum relation

E^2 = p^2c^2 + m^2c^4

Total energy contains both rest energy and the contribution associated with momentum. The relation also applies to massless particles: a photon has energy and momentum even though it has no rest mass. OpenStax · Energy and momentum in relativity

Relativistic energy–momentum relation: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$E$	Total energy in the chosen inertial frame	$\mathrm{J}$	—
$p$	Magnitude of momentum in that frame	$\mathrm{kg\,m\,s^{-1}}$	—
$m$	Invariant mass	$\mathrm{kg}$	—
$c$	Speed of light in vacuum	$\mathrm{m\,s^{-1}}$	$299\,792\,458$

When it applies: A special-relativistic relation for a free particle or a system’s total energy, total momentum, and invariant mass. It does not treat mass as increasing with speed.

Time dilation

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

A moving clock accumulates less time between two events than the coordinate time assigned by an observer in an inertial frame. The difference is tiny at everyday speeds but measurable for fast particles and precision clocks. OpenStax · Time dilation

Time dilation: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\Delta t$	Elapsed coordinate time in the observer’s inertial frame	$\mathrm{s}$	—
$\Delta\tau$	Proper time measured by the moving clock between the same events	$\mathrm{s}$	—
$v$	Clock’s speed relative to the observer	$\mathrm{m\,s^{-1}}$	—
$c$	Speed of light in vacuum	$\mathrm{m\,s^{-1}}$	$299\,792\,458$

When it applies: This form assumes constant relative speed below light speed in flat spacetime. Varying motion requires integration, and gravity introduces additional time effects.

Einstein’s field equations

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}

Spacetime geometry is linked to the distribution of energy, momentum, and stress. These equations underpin the description of black holes, gravitational waves, and cosmic expansion. The compact notation represents a coupled set of equations rather than a single arithmetic formula. Sean Carroll · Field equations and the cosmological constant

Einstein’s field equations: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$G_{\mu\nu}$	Einstein tensor, built from spacetime curvature	$\mathrm{m^{-2}}$	—
$g_{\mu\nu}$	Metric tensor, describing spacetime intervals	$1$	—
$\Lambda$	Cosmological constant	$\mathrm{m^{-2}}$	—
$T_{\mu\nu}$	Stress–energy tensor, describing energy, momentum, and stresses	$\mathrm{J\,m^{-3}}$	—
$G$	Newtonian gravitational constant	$\mathrm{m^3\,kg^{-1}\,s^{-2}}$	$\approx 6.67430\times10^{-11}$
$c$	Speed of light in vacuum	$\mathrm{m\,s^{-1}}$	$299\,792\,458$
$\mu,\ \nu$	Indices labelling spacetime components	—	—
$\pi$	Ratio of a circle’s circumference to its diameter	$1$	$\approx 3.141592653589793$

Tensor units assume coordinates measured in metres, including light speed multiplied by time for the time coordinate, and a dimensionless metric. Other coordinate conventions can change component units. The cosmological constant is model-dependent, with no universal numerical value.

When it applies: The classical theory of gravity, written here with SI constants. Tensor calculus, a matter model, and initial or boundary data are needed to solve it; quantum gravity lies beyond this description.

Quantum physics

Energy quanta, matter waves, probability amplitudes, and the limits on jointly sharp measurements.

Photon energy

E = hf

Light exchanges energy in quanta called photons. A photon’s energy depends on frequency, so ultraviolet photons carry more energy than visible or infrared photons. Increasing intensity at a fixed frequency increases photon number rather than energy per photon. OpenStax · Photons and the photoelectric effect

Photon energy: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$E$	Energy of one photon	$\mathrm{J}$	—
$h$	Planck constant	$\mathrm{J\,s}$	$6.62607015\times10^{-34}$
$f$	Light frequency	$\mathrm{Hz}$	—

When it applies: Describes energy per photon at a given frequency. A light beam can contain many photons and, for broadband light, many frequencies.

De Broglie wavelength

\lambda = \frac{h}{p}

Matter has wave properties as well as particle properties. A particle with larger momentum has a shorter wavelength. Electron diffraction reveals this behaviour, while the wavelengths of ordinary moving objects are usually too small to notice. OpenStax · De Broglie’s matter waves

De Broglie wavelength: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\lambda$	Matter-wave wavelength	$\mathrm{m}$	—
$h$	Planck constant	$\mathrm{J\,s}$	$6.62607015\times10^{-34}$
$p$	Magnitude of the particle’s momentum	$\mathrm{kg\,m\,s^{-1}}$	—

When it applies: Applies directly to a momentum component of a free particle’s state. A localised wave packet contains a range of momenta and wavelengths.

Time-dependent Schrödinger equation

i\hbar\frac{\partial\psi}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2 + V\right]\psi

The wavefunction evolves according to the particle’s kinetic and potential energies. Solving this equation with suitable conditions predicts quantum states and how they change over time. It describes an amplitude whose squared magnitude determines position probabilities, rather than a classical trajectory. OpenStax · The Schrödinger equation

Time-dependent Schrödinger equation: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\psi$	Wavefunction, depending on position and time	$\mathrm{m^{-3/2}}$	—
$i$	Imaginary unit	$1$	$\sqrt{-1}$
$\hbar$	Reduced Planck constant	$\mathrm{J\,s}$	$\approx 1.054571817\times10^{-34}$
$m$	Particle mass	$\mathrm{kg}$	—
$V$	Potential energy, possibly varying with position and time	$\mathrm{J}$	—
$\nabla^2$	Laplacian: sum of second spatial derivatives	$\mathrm{m^{-2}}$	—
$\partial/\partial t$	Time derivative at fixed position	$\mathrm{s^{-1}}$	—

Wavefunction units assume normalisation over three-dimensional position space. In one or two dimensions, the normalisation gives different units.

When it applies: This is the nonrelativistic single-particle form for a scalar potential, with spin and magnetic vector potentials omitted. Relativistic particles need a different equation.

Born rule for position

P(\mathbf{r}\in\mathcal{R}) = \int_{\mathcal{R}} |\psi(\mathbf{r},t)|^2\,d^3r

The squared magnitude of the wavefunction is a position probability density. Integrating it over a region gives the probability that a position measurement finds the particle there. This connects the mathematical quantum state to observable outcomes. OpenStax · Wavefunctions and probability

Born rule for position: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$P$	Probability of finding the particle in the chosen region	$1$	—
$\mathbf{r}$	Position vector	$\mathrm{m}$	—
$t$	Time	$\mathrm{s}$	—
$\mathcal{R}$	Region of space being measured	—	—
$\psi$	Normalised wavefunction	$\mathrm{m^{-3/2}}$	—
$d^3r$	Volume element in three-dimensional space	$\mathrm{m^3}$	—

When it applies: For a single particle in a pure state with the wavefunction normalised over all space. The total position probability is one.

Heisenberg uncertainty principle

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

A quantum state cannot have both perfectly sharp position and perfectly sharp momentum along the same axis. Narrowing the position distribution broadens the range of momenta. This is a property of quantum states, not simply a limit caused by poor measuring equipment. OpenStax · Heisenberg uncertainty principle

Heisenberg uncertainty principle: symbols, definitions, SI units, and constant values
Symbol	Meaning	SI unit	Constant value
$\Delta x$	Standard deviation of position measurements along one axis	$\mathrm{m}$	—
$\Delta p_x$	Standard deviation of momentum measurements along that same axis	$\mathrm{kg\,m\,s^{-1}}$	—
$\hbar$	Reduced Planck constant	$\mathrm{J\,s}$	$\approx 1.054571817\times10^{-34}$

When it applies: The spreads refer to distributions for identically prepared states, not a classical measurement’s error bars. The inequality gives a lower bound, not a mandatory equality.