Astrophysics
Kepler's Laws of Planetary Motion
Kepler's three laws describe planetary motion around the Sun: orbits are ellipses with the Sun at one focus, planets sweep equal areas in equal times, and the square of orbital period is proportional to the cube of the semi-major axis. These empirical laws provided the foundation for Newton's law of gravitation and remain fundamental to celestial mechanics and orbital dynamics.
Vis-Viva Equation
The Vis-Viva equation relates the velocity of a particle in an elliptical, hyperbolic or parabolic orbit to the distance to the barycenter, mass of the body it's orbiting, and the semi-major axis of the orbit. This sets the basis for velocity calculations in non-circular orbits and allows us to later derive equations like the equation for escape velocity with ease.
Classical Physics
Angular Momentum and Torque
Angular momentum is the rotational analog of linear momentum, quantifying the rotational motion of a particle or system about a reference point. Defined as the cross product of position and momentum vectors, it is an axial vector whose direction indicates the rotation axis.
🤖 DRAFTMaxwell Equations
Maxwell's equations form the foundation of classical electromagnetism, describing how electric and magnetic fields interact with charges and currents. These four coupled partial differential equations unify electricity and magnetism into a single electromagnetic theory, predicting the existence of electromagnetic waves including light.
Principle of Least Action
The principle of least action states that the path taken by a physical system between two configurations is the one that makes the action stationary. The action is defined as the time integral of the Lagrangian function.
Speed of Light from Maxwell's Equations
In vacuum, Maxwell's equations imply that disturbances of the electromagnetic field propagate as waves whose speed is fixed by the electric permittivity and magnetic permeability of free space. Identifying this speed with the experimentally measured speed of light shows that light is an electromagnetic wave and sets the value of the universal constant `c` from purely electric and magnetic static measurements such as the force between charges or currents.
General Relativity and Quantum Cosmology
Gravitational Field and Potential
The gravitational field describes the gravitational influence that a massive body extends into the space around it. Defined as the force per unit mass experienced by a test particle, the field provides a local characterization of gravitational effects.
🤖 DRAFTRelativistic Energy and Momentum
Einstein's special relativity revolutionizes our understanding of energy and momentum at high velocities. By redefining force as the rate of change of relativistic momentum and calculating the work done to accelerate a particle, the theory derives the famous relativistic energy formula E = γmc².
Special Relativity Coordinate Transformations
Einstein's special theory of relativity derives the coordinate transformations between inertial reference frames moving at constant relative velocity. Built on two fundamental postulates - the relativity principle and the constancy of light speed - the theory reveals that space and time are unified into spacetime.
High Energy Physics (Theory)
Canonical Quantization of the Free Dirac Field
Canonical quantization of the Dirac field promotes the classical spinor field to an operator satisfying anticommutation relations, as required by the spin-statistics theorem for spin-1/2 fermions. The field is expanded in plane-wave modes with positive-frequency terms associated with particle annihilation operators and negative-frequency terms with antiparticle creation operators.
🤖 DRAFTCanonical Quantization of the Free Scalar Field
Starting from the Klein-Gordon field and assuming canonical commutation relations, the mode operator algebra is derived and the Fock space of particle states is constructed. The ladder algebra of the number operator shows that mode operators raise or lower particle number, justifying the names creation and annihilation operators.
Dirac Equation
The Dirac equation is the relativistic wave equation for spin-1/2 fermions. It resolves the challenge of creating a quantum equation that is first-order in time, ensuring positive probability density, while being consistent with special relativity.
Fock Space
Fock space is the Hilbert space for systems with variable particle number, fundamental to quantum field theory. It is constructed as the direct sum of n-particle Hilbert spaces, starting from the vacuum which contains no particles.
Klein-Gordon Lagrangian and Hamiltonian Density
The Klein-Gordon Lagrangian density provides the classical field theory foundation for the relativistic scalar field. From this Lagrangian, the Klein-Gordon equation emerges via the Euler-Lagrange field equations.
Lorentz group and four-vectors
The Lorentz group is the set of linear transformations that preserve the Minkowski metric and therefore leave the spacetime interval invariant. A four-vector is any object whose components transform with a Lorentz matrix, ensuring that Minkowski inner products are frame-independent.
Spin–Statistics Theorem
The spin–statistics theorem connects intrinsic spin with exchange symmetry of identical particles. Integer-spin particles have symmetric many-particle states under exchange, while half-integer-spin particles have antisymmetric states.
Mathematical Physics
Quantum Physics
Blackbody Radiation
Planck's law describes the spectral energy density of black-body radiation as a function of frequency and temperature. By quantizing electromagnetic modes with energy quanta `h*nu`, it resolves the ultraviolet catastrophe of classical physics and underpins modern quantum theory and thermal emission models.
Born Rule
The Born rule specifies how a quantum state yields measurable probabilities for any well-defined measurement. Given a state and a measurement description, it assigns a normalized probability to each possible outcome in a way consistent with observed frequencies under repeated trials.
Klein-Gordon Equation
The Klein-Gordon equation is the relativistic wave equation for spin-0 (scalar) particles. It arises from applying quantum mechanical operator substitutions to the relativistic energy-momentum relation `E^2 = (p*c)^2 + (m*c^2)^2`.
Ladder Operators for the Quantum Harmonic Oscillator
Ladder operators provide an algebraic method to solve the quantum harmonic oscillator without directly solving differential equations. The annihilation operator `a` lowers the energy eigenstate by one quantum, while the creation operator `a^(**)` raises it.
Schrödinger Equation
The Schrödinger equation gives the time evolution of the quantum state of a nonrelativistic particle. In its time-dependent form it is a linear first-order equation in time relating the complex wavefunction `psi(bbr,t)` to the Hamiltonian operator built from kinetic and potential energy.
Uncertainty Principle
The Heisenberg uncertainty principle states that for any pair of non-commuting observables their statistical spreads (standard deviations) `Delta A` and `Delta B` in a given quantum state cannot both be made arbitrarily small. The general Robertson relation ties the product `Delta A * Delta B` to the expectation value of their commutator, while canonical coordinate-momentum pairs `(q_i, p_i)` satisfy `Delta q_i * Delta p_i >= hbar/2`.