How physics work - the roles of postulates, experiments, and logical derivation

Physics connects mathematical models with observations. To understand a result, identify what was assumed, what the symbols represent, what was deduced, and what was measured. A correct proof establishes a consequence of its premises; applying that consequence to a real system also requires evidence that the premises describe it adequately.

What a physical model contains

A theory supplies a general framework, such as classical mechanics or quantum mechanics. A model selects a mathematical representation and physical premises for a particular class of systems. It can draw on several theories, provided their approximations and coupling are compatible. Models need more than a list of laws: their quantities, interpretation, conditions and scope must be specified. Models in Science discusses these different uses of models.

Interpretation and assumption have different jobs. Calling $x$ displacement gives it physical meaning; assuming negligible damping claims that an omitted effect is small enough for the intended application. An idealisation sets that effect exactly to zero in the model. An approximation concerns how closely the resulting prediction represents the system.

Postulates and other physical premises

Physical premise is a useful umbrella term. A postulate is a principle adopted within a framework or model. An empirical law records a regularity inferred from observations. Measured parameter values, calibration relations, and initial or boundary conditions can also enter a deduction. Calling every input a postulate conceals these distinctions.

Experiments may motivate a general law, but the move from finite observations to a general postulate includes inference and judgement. Some postulates have direct experimental support; others are tested through their consequences or adopted provisionally. Idealising assumptions, such as perfectly rigid walls, need not be literally realised. Their value depends on the predictions and regimes they enable.

Mathematical theorems and physics theorems

A mathematical theorem follows from mathematical axioms, definitions and established results. A physics theorem additionally interprets its quantities physically and may use physical premises. Its deduction must meet the same logical standard, while its application remains conditional on those premises.

For the spring, take $m,k>0$ constant and assume

$m\ddot{x}=-kx\mathrm{.}$

A dot denotes differentiation with respect to time. Define

$E(t)=\frac{1}{2}m\dot{x}(t{)}^2+\frac{1}{2}kx(t{)}^2\mathrm{.}$

The chain rule gives

$\dot{E}=\dot{x}(m\ddot{x}+kx)=0.$

Thus $E$ is constant on the time interval. This is an exact mathematical consequence of the stated equation. Interpreting it as mechanical energy conservation for a real spring requires the physical meaning of $m,k,x$ and the assumptions about force, damping and forcing.

Mathematical physics also develops the structures, existence results and approximation methods underlying physical theories. Deriving their consequences is a central activity, but does not exhaust the subject.

Experiments supply inputs and test predictions

Experiments connect the model to observations in several ways:

• Construction: observations motivate or constrain a law or modelling assumption.
• Calibration and preparation: measurements supply parameters, initial conditions or boundary data.
• Operational interpretation: an apparatus and procedure specify how a quantity is measured.
• Testing: observed outcomes are compared with predictions made under stated conditions.

A model can include a mathematical description of its measurement apparatus. The laboratory procedure and its recorded outcomes remain empirical activities and data. Experimental results can become premises of a deduction; carrying out an experiment is not itself a logical inference rule.

An admitted experimental premise therefore plays a premise-like role analogous to a mathematical axiom within that argument. Its evidential basis differs: measurements have uncertainty and depend on preparation, calibration and analysis. Exact algebra does not remove that uncertainty. NIST's guidance explains how measurement results express it.

For example, substituting $x(t)=A\cos (\omega t+\varphi )$, with nonzero amplitude $A$ and phase $\varphi$, into the spring equation gives ${\omega }^2=k\mathrm{/}m$ and period $T=2\pi \sqrt{m\mathrm{/}k}$. If a justified input bound is $0<k_{\min }\le k\le k_{\max }$, with $m$ fixed in the model, monotonicity gives

$2\pi \sqrt{\frac{m}{k_{\max }}}\le T\le 2\pi \sqrt{\frac{m}{k_{\min }}}\mathrm{.}$

The deduction is exact conditional on that bound. A reported standard uncertainty is not automatically a guaranteed interval; its statistical interpretation must be retained.

What agreement with experiment establishes

Testing assesses agreement within specified uncertainty, conditions and observables. It can support a model, reveal limitations, or favour alternatives. A discrepancy tests the model together with its preparation, apparatus and analysis assumptions; it does not automatically identify which premise failed.

Acceptance is an evidence-informed judgement about reliability and applicability. Predictive accuracy, reproducibility, uncertainty, consistency and competing explanations constrain it. It is neither an arbitrary preference nor a deductive proof of universal truth. A model calibrated on a dataset matching that dataset is also a different achievement from predicting independent observations.

Deterministic and probabilistic predictions

Determinism concerns a model: when its equations and complete initial data determine a unique evolution, its prediction is deterministic. Standard quantum mechanics includes deterministic unitary state evolution between measurements and probabilistic predictions for measurement outcomes. Wave functions and their probability interpretation describe this second role.

If a prepared spin state has equal amplitudes for the two outcomes of a specified measurement, the Born rule assigns each probability $1\mathrm{/}2$. That predicts probabilities, rather than selecting the next outcome. For $N$ independent repetitions, the expected count of either outcome is $N\mathrm{/}2$, but a finite sample need not split evenly.

These different predictions do not make reality ill-defined. The distinction is between formal consequences of a representation and actual systems and observations. Comparing those consequences with observations does not, by itself, settle whether nature is fundamentally deterministic or random.

Two exact models and their physical limits

Classical mechanics: a projectile

Assume constant downward acceleration $g>0$, no drag, and initial position $(0,y_0)$ with velocity $(u,w)$, where $u\mathrm{\ne }0$. Integrating $\ddot{x}=0$ and $\ddot{y}=-g$ gives

$x(t)=ut,y(t)=y_0+wt-\frac{1}{2}g{t}^2\mathrm{.}$

Eliminating $t=x\mathrm{/}u$ yields the exact parabola

$y(x)=y_0+\frac{w}{u}x-\frac{g}{2u^2}{x}^2\mathrm{.}$

The calculation proves the trajectory in this model. Drag, wind, spin and varying gravity can make a real trajectory differ. The projectile-motion model is useful where those effects are sufficiently small for the desired accuracy.

Quantum mechanics: an infinite well

The one-dimensional infinite-well model confines a particle of mass $m>0$ to $0<x<L$, with wavefunction zero outside and boundary conditions $\psi (0)=\psi (L)=0$. Inside, the stationary Schrödinger equation is

$-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\mathrm{\prime }\mathrm{\prime }}=E\psi ,$

where $\mathrm{\hslash }$ is the reduced Planck constant and $E$ an energy eigenvalue. For a nonzero state, the boundary conditions select $\psi (x)=A\sin (n\pi x\mathrm{/}L)$, giving

$E_n=\frac{n^2{\pi }^2{\mathrm{\hslash }}^2}{2m{L}^2},n=1,2,\dots \mathrm{.}$

The energy eigenvalues and zero probability outside are exact within the infinite-well model. Finite barriers can permit penetration into the surrounding region and change the energy levels. The ideal boundary condition does not establish that a physical device has infinite walls.

Combining frameworks: molecules and chemical reactions

A molecular model may treat nuclear motion classically while obtaining electronic energies from quantum mechanics. The electrostatic pair energy

$\frac{q_i{q}_j}{4\pi {\epsilon }_0{r}_{ij}}$

can enter a quantum Hamiltonian: $q_i,q_j$ are charges, $r_{ij}$ their separation, and ${\epsilon }_0$ the vacuum permittivity. Electromagnetic interactions underlie bonding, but covalent bonds require a quantum description of electronic states and their symmetry. They are not explained merely by assigning classical attractive forces between electrons and nuclei. See molecular bonding.

Such a combined model must specify which degrees of freedom receive each description and when its approximations apply. It does not inherit every postulate of every framework indiscriminately.

Collisions can transfer kinetic energy into internal motion and bond rearrangement. Collision energy, orientation and a reaction's activation barrier matter; “enough force” alone does not specify whether a reaction occurs. Collision theory explains these conditions.

Breaking an isolated bond requires energy. Forming bonds can release energy. The net reaction balance determines whether energy is released to the surroundings; breaking bonds alone does not produce heat. At constant pressure, an exothermic reaction has negative reaction enthalpy under the specified conditions. Released energy can populate molecular motion and internal modes, then be redistributed by collisions. The bond-energy account also explains the limits of estimating reaction enthalpies from average bond energies.

Heat, temperature and collision frequency

Internal energy is the system's microscopic kinetic and interaction energy. Heat is energy transferred because of a temperature difference; it is not a substance stored in a body. Temperature is a thermodynamic state variable. In a classical ideal gas at equilibrium, it relates to average translational kinetic energy by

$⟨K_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}⟩=\frac{3}{2}{k}_{B}T,$

where $k_B$ is Boltzmann's constant and $T$ absolute temperature. This scoped relation is not a universal definition of temperature for every material. See kinetic theory and thermodynamic systems.

Collision frequency depends on density, relative speeds and collision cross-sections. It influences energy redistribution and reaction rates, but is not itself heat or temperature. Increasing the number of ideal-gas molecules at fixed temperature increases total internal energy, without increasing their average translational energy. More frequent collisions do not compensate for lower average kinetic energy to make that equilibrium gas hotter.

Making the distinctions explicit when learning physics

For each calculation, ask:

1. Which framework, laws and mathematical objects are being used?
2. What do the symbols mean physically, and how are they measured?
3. Which inputs are measured, assumed or idealised?
4. What follows deductively, including uncertainty inherited from the inputs?
5. Which observations test the prediction, and in what regime?

This separates solving a model from judging its applicability. It also explains how a rigorous deduction can coexist with uncertain inputs, limited empirical support, and useful approximations.