Principles
Canonical Commutation Relations
Canonical variables `q_i` and `p_i`, representing generalized coordinates and their conjugate momenta for each degree of freedom, satisfy the commutation relations `[q_i, p_j] = i*ℏ*delta_(ij) `, `[q_i, q_j] = 0`, and `[p_i, p_j] = 0`. These relations encode the basic quantum kinematics of canonical pairs and fix the fundamental scale through `ℏ`.
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Classical Electromagnetism Framework
Fields and sources are classical, smooth (continuously differentiable, `C^1` or better), and defined on flat Minkowski spacetime. Vector calculus theorems (divergence theorem, Stokes' theorem) apply. Quantum and gravitational effects are negligible.
Conservation Laws
Total energy, linear momentum and angular momentum are conserved.
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Correspondence Principle
Quantum mechanics must reproduce classical mechanics in the appropriate limit (large quantum numbers, macroscopic scales, or formally `ℏ -> 0`). In particular, the quantum Hamiltonian operator is chosen so that its spectrum and dynamics reduce to those of the classical Hamiltonian `H_("classical")(p, r, t)` when quantum effects become negligible.
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Electromagnetic Field Quantization
The electromagnetic field is quantized, with each mode having discrete energy levels. Each mode of frequency `nu` can only have energies that are integer multiples of `h*nu`, where `h` is Planck's constant and the integer represents the photon occupation number.
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Gleason Theorem Conditions
The Hilbert space has dimension at least 3 (`dim(H) >= 3`), and the probability assignment on projectors (orthogonal projection operators `P` satisfying `P^2 = P = P^ast`) is continuous
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Hilbert Space Probability Structure
Quantum measurement outcomes correspond to orthogonal projection operators (projectors) on a complex Hilbert space `H`, where projectors are self-adjoint operators satisfying `P^2 = P = P^ast`. Probabilities can be assigned to these projectors via a probability measure `mu` mapping projectors to real numbers in `[0,1]`.
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Inertial Reference Frame
Observations are made from an inertial (non-accelerating) reference frame
Planck-de Broglie Relations
Energy and momentum of matter waves are related to frequency and wavevector by `E = ℏ*omega` and `p = ℏ*k`.
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Quantum Observables as Self-Adjoint Operators
Each physical observable is represented by a self-adjoint (Hermitian) operator `A` on the Hilbert space. The possible measurement values are identified with the (real) spectrum of `A`. For a normalized state `|:psi:)`, the quantity `a_0 = (:psi:|A|:psi:)` defines the expectation value of `A` in that state and is real.
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Quantum Superposition and Linear Dynamics
Quantum states form a complex vector space and physically allowed time evolutions act linearly on this space. If `psi_1` and `psi_2` are solutions of the evolution equation, any complex linear combination `alpha*psi_1 + beta*psi_2` is also a solution.
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Stationary Action Principle
The system obeys the principle of stationary action (least action principle), meaning the path taken by the system makes the action stationary, leading to the Euler-Lagrange equations
Variational Calculus Framework
System describable by generalized coordinates `q_i(t)` with well-defined, twice-differentiable Lagrangian `L(q_i, dot q_i, t)`, smooth trajectories, and suitable boundary conditions for variational analysis (fixed endpoints, allowable interior variations)
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Empirical
Ampère-Biot-Savart Law
Steady electric currents generate magnetic fields that circulate around them. The line integral of magnetic field around a closed loop equals `mu_0` times the enclosed current.
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Coulomb's Law
Electric charges interact via an inverse-square force law. A point charge `q` produces an electric field `E(r) = q / (4 * pi * epsilon_0 * r^2) * hat(r)` where `hat(r)` is the radial unit vector.
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Electromagnetic Polarization
Electromagnetic waves have two independent polarization directions (transverse to propagation direction)
Faraday's Law of Electromagnetic Induction
A time-varying magnetic flux through a closed circuit induces an electromotive force (emf) in that circuit. The induced emf equals the negative rate of change of magnetic flux.
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Light Speed Constant
The speed of light in vacuum is constant and independent of the motion of the source or observer
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No Magnetic Monopoles
Magnetic monopoles (isolated magnetic charges) have never been observed experimentally. As a consequence, the net magnetic flux through any closed surface is zero. (no free magnetic charge density exists).
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Well-Defined Rest Mass
Particles have well-defined rest masses that do not depend on the reference frame
Approximations
Classical Macroscopic Limit
For macroscopic bodies with de Broglie wavelength `λ_(dB) < < L`, quantum effects such as interference and wave-particle duality are negligible and the motion can be described by classical trajectories.
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Constant Relative Velocity
Reference frames move with constant relative velocity (no acceleration)
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Flat Spacetime
Spacetime is flat, no gravitational effects considered
Linear Isotropic Vacuum
The medium is vacuum (free space) characterized by constants `epsilon_0` (electric permittivity) and `mu_0` (magnetic permeability), but may contain charge density `rho(r,t)` and current density `J(r,t)`. The vacuum is linear, homogeneous, and isotropic with constitutive relations `D = epsilon_0 * E` and `H = B / mu_0`.
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Nonrelativistic Regime
Particle velocities satisfy `v < < c` so that relativistic corrections to the kinetic energy and dynamics can be neglected. In this limit the kinetic energy is well approximated by `E ≃ p^2/(2*m)`.
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Perfect Blackbody
The spectral energy density `u(nu,T)` describes radiation from a perfect black body - an idealized emitter that absorbs all incident radiation and re-emits it purely based on temperature `T`
Point Mass Approximation
Both bodies can be treated as point masses with spherically symmetric mass distributions
System Isolation
System is sufficiently isolated from external influences (for gravitational systems: negligible external perturbations)
Thermal Equilibrium
The radiation field is in thermal equilibrium at temperature `T`, meaning the emission and absorption rates are balanced and the energy distribution is time-independent