Math-and-Physics-How-To

A repository of publicly-accessible resources for learning & self-teaching mathematics & physics up to the graduate level.

Scroll to the bottom of the page to see Onri's Table of Critical Equations

"Being good or great at mathematics is a matter of practice. Isolating a term in a complex mathematical formula is like untangling a knot, it requires persistence & loosening of things". - O.J.B.

Physics <-- Greek phýsis (“nature”) --> “study of what nature does.”

Parameter <-- Greek para (“beside”) + metron (“measure”) --> “a measurable beside the main variables.”

Stimulus <-- Latin stimulus (“goad, spur”). Response <-- Latin respondēre (“answer”).

Free Mathematics Textbooks Resources:

In Case You Want Some Fun Open Access Interactive Tools to Try:

Free Education

Great Resources for Practicing Computational Methods:

Map of Mathematics & Their Prerequisites:

graph TD
    A[Start Learning Mathematics]
    B[Logic]
    C[Sets and Maps]
    D[Numbers]
    E[Real Numbers]
    F[Complex Numbers]
    G[Real Analysis]
    H[Linear Algebra]
    I[Algebra]
    J[Abstract Linear Algebra]
    K[Multivariable Calculus]
    L[Ordinary Differential Equations]
    M[Complex Analysis]
    N[Functional Analysis]
    O[Measure Theory]
    P[Multidimensional Integration]
    Q[Probability Theory]
    R[Fourier Transform]
    S[Unbounded Operators]
    T[Hilbert Spaces]
    U[Distributions]
    V[Manifolds]

    A --> B
    A --> C
    A --> D
    B --> G
    C --> G
    D --> E
    D --> F
    E --> G
    F --> M
    G --> H
    H --> G
    G --> K
    G --> L
    G --> M
    H --> I
    H --> J
    H --> K
    K --> N
    K --> O
    K --> P
    L --> Q
    M --> R
    N --> S
    N --> T
    N --> U
    O --> Q
    P --> Q
    P --> V
    Q --> V
    R --> Q

    %% Group 1: Formerly Yellow (Foundation) - Now Medium Blue
    style B fill:#2E64FE,stroke:#153E7E,stroke-width:2px,color:#FFFFFF
    style C fill:#2E64FE,stroke:#153E7E,stroke-width:2px,color:#FFFFFF
    style D fill:#2E64FE,stroke:#153E7E,stroke-width:2px,color:#FFFFFF
    style E fill:#2E64FE,stroke:#153E7E,stroke-width:2px,color:#FFFFFF
    style F fill:#2E64FE,stroke:#153E7E,stroke-width:2px,color:#FFFFFF

    %% Group 2: Formerly Light Green (Core) - Now Dark Blue
    style A fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF
    style G fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF
    style H fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF
    style I fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF
    style J fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF
    style K fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF
    style L fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF
    style M fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF

    %% Group 3: Formerly Dark Green (Advanced) - Now Very Dark/Navy Blue
    style N fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF
    style O fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF
    style P fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF
    style Q fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF
    style R fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF
    style S fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF
    style T fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF
    style U fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF
    style V fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF

Adapted from: Learn with the Map of Mathematics, The Bright Side of Mathematics (2023)
https://youtu.be/ljGSId-uHw8?si=xKNup3hOVsWC6uTv&t=200

Strategic Approach to Solving Mathematical, Physical, & Engineering Problems Manually by (O.J.B.):

graph TD
    A[Highlight Key Terms]
    B[Record Linguistic Definitions]
    C[Locate Required Equations]
    D[Convert Units & Formulas]
    E[List or Tabulate Assumed/Ignored Terms]
    F[Script into LaTeX/Markdown]
    G[Import the LaTeX Script in Google Colab]
    H[Solve Problems Sequentially with Python]
    I[Double-check with Textbooks]
    J[Validate with Research Literature]

    %% Connections
    A --> B
    A --> C
    C --> D
    C --> E
    E --> F
    F --> G
    G --> H
    H --> I
    H --> J

    %% Define Blue Classes (Medium, Dark, Navy)
    classDef mediumBlue fill:#2E64FE,stroke:#153E7E,stroke-width:2px,color:#FFFFFF
    classDef darkBlue fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF
    classDef navyBlue fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF

    %% Apply Styles based on flow depth
    %% Top Level: Initial Analysis
    class A,B,C mediumBlue
    
    %% Middle Level: Formatting & Setup
    class D,E,F,G darkBlue
    
    %% Bottom Level: Execution & Verification
    class H,I,J navyBlue

A Technique For Training a Large Language Model Strategically on Real Mathematics Resources by (O.J.B.)

graph TD
    A[Start: Locate Literature Online] -- Search forums & textbooks --> B[Identify Accessible PDFs]
    B --> C{Comprehensible?}
    C -- No --> D(Discard)
    C -- Yes --> E[Add to List or Tree]
    E -- Repeat Process --> E
    E --> F[Compile Best Literature Set]
    
    subgraph Optional
    F -.-> G[Repeat for Dissertations]
    end
    
    F --> H[Save all PDFs in One Folder]
    G --> H
    H -- Select Top 10 --> I[Train LLM on PDFs]
    I --> J[LLM Identifies Math Patterns]
    J --> K[Generate LaTeX/Markdown Scripts]
    K --> L[Find Supporting Literature]
    L --> M((End: Verify Open Access))

    %% Define Blue Classes (Medium, Dark, Navy)
    classDef mediumBlue fill:#2E64FE,stroke:#153E7E,stroke-width:2px,color:#FFFFFF
    classDef darkBlue fill:#0033CC,stroke:#001F7A,stroke-width:2px,color:#FFFFFF
    classDef navyBlue fill:#000066,stroke:#000033,stroke-width:2px,color:#FFFFFF

    %% Apply Styles
    %% Phase 1: Search & Filter (Medium Blue)
    class A,B,C,D,G mediumBlue
    
    %% Phase 2: Organization (Dark Blue)
    class E,F,H darkBlue
    
    %% Phase 3: AI Processing & Output (Navy Blue)
    class I,J,K,L,M navyBlue

Pro Tips for Solving Mathematical Problems by (O.J.B.):

Tip	Explanation/ Details
Substitute given variables with custom or other existing variables (e.g., AbcdEfG)	Solve the equation using your own variables, then mirror the steps onto the original problem for proportional reasoning. A technique to solving a formula is to find a term that exists in a different equation, followed by solving for that desired term (isolating it to one side).
Interpret the equal sign as "converts to"	Thinking of "=" as "converts to" can facilitate substitutions & manipulations in other mathematical expressions.
Think in terms of ratios by default	Viewing values as ratios can simplify problem-solving & conceptual understanding.
Understand the difference between analytical vs. numerical approaches	Exact solutions are often associated with analytical approaches while approximation or discretized solutions are often associated with numerical appoaches.
Isolating a term in a complex mathematical formula is like untangling a knot	Sometimes one needs to loosen things before making real progress, and every move must be made with careful consideration of how it affects the whole structure. Being well-organized & systematic can take one a very long way in problem solving.
Use software tools for conversion to markdown or LaTeX	Convert equations for better inspection & rendering, ensuring accuracy.
Leverage Python & libraries like SymPy	Write equations in Python for execution & manipulation, aiding clarity & verification.
Remember solutions on graphs are line intersections	Graphical solutions typically correspond to intersection points of lines or curves.
It is safe to assume invisible exponents of 1 as well as invisible grouping symbols	Keeping this in mind helps to maintain organization for obtaining a correct result. It also helps with being able to linearize the writing format for code and things like LaTeX.
It is useful to think of formulas as having grouped components within invisible grouping symbols	In such a case, a grouped component may likely be acting like a modulator or scaling factor.
Use preferred mathematical notations	Include curly brackets, e-notation, prime/dot notation, & highlight invisible symbols for clarity & precision.
Stay aware of term replacements	Recognize when terms are replaced or approximated in mathematical contexts. Keywords: replacement, approximation.
Consider various methods (axiomatic, first principles, empirical)	Use diverse approaches, including logical derivations, empirical evidence, & hybrid methodologies for problem-solving.
Explore graphical, tabulated, or geometric representations	Visual or tabular methods can simplify complex mathematical concepts.
Practice final exam reviews	Mastery in mathematics comes with regular & extensive practice, particularly of exam-style problems.

Table of Useful General Assumptions for Abstract Terminology Used in Everyday Life:

Term or Concept	Description or Note
Another way of saying abstract	“Indeterminate” or “boundless".
Light	Generally refers to the electromagnetic field or electromagnetic radiation. When quantized, it usually refers to photons but can also mean quantized modes, coherent and squeezed states, polaritons, or plasmons and surface plasmon polaritons.
Optics	A branch of physics concerned with the generation, propagation, manipulation, and detection of electromagnetic radiation (especially in and around the visible range), as well as its interactions with matter, including phenomena like reflection, refraction, diffraction, interference, and polarization. There are sub-disciplines or sub-classifications of optics such as electron, ion, and quantum optics.
Plastic	Often used metaphorically to refer to something moldable or flexible; in a physical context, it can mean a polymer material or exhibit plastic (irreversible) deformation.
Radiation	The emission or transmission of energy through space or a medium in the form of electromagnetic waves (e.g., radio waves, visible light, X-rays, gamma rays) or subatomic particles (e.g., alpha, beta, neutrons). In a broader sense, it can also refer to acoustic waves, though in physics “radiation” typically implies electromagnetic or particle radiation. Many people equate “radiation” solely with ionizing radiation, which is harmful in large doses and includes X-rays, gamma rays, and high-energy particles that can ionize atoms. This is a common misunderstanding because non-ionizing radiation (like visible light, microwaves, radio waves) is also “radiation,” just not ionizing.
Electric field	Generally refers to the force per unit charge in a region due to a static or dynamic charge distribution. It can also be generated or excited by time-varying magnetic fields (as in electromagnetic waves) or by charge redistribution effects caused by incident radiation (e.g., in the photoelectric effect). Note: In formal treatments, the photoelectric effect is more often framed in a quantum context rather than purely classical terms.
Pure electric fields can be justified as being “pure”	If they originate from idealized situations: a point charge, a parallel plate capacitor, uniformly charged conductors, electrostatic lenses (vacuum tube-based focusing), and electric dipole fields.
Space	A boundless, continuous, or discrete extent in which objects, fields, or systems exist, interact, and have relative positions. In STEM, it can refer to physical space (three-dimensional Euclidean or non-Euclidean structures), mathematical spaces (vector spaces, phase spaces, Hilbert spaces, and even two-dimensional planes such as Cartesian planes or complex planes), or conceptual spaces (design spaces, information spaces). In everyday life, it denotes an area, room, or capacity for existence, movement, or thought.
Platform(s)	Can be a physical or conceptual foundation upon which systems, processes, or experiments are built.
Linearity	1:1 ratio in the response by default, with a consistent or zero rate of change.
Non-linearity	Any relationship where the ratio or response exhibits a non-zero rate of change; not a simple 1:1 linear relationship.
Theory	Often confused with “hypothesis” or “mathematical justification.” In science, a theory is a well-substantiated explanation of some aspect of the natural world.
Magnetic	Refers to magnetism. By default, all matter made of atoms is at least diamagnetic. If it is not diamagnetic, its magnetism (paramagnetism, ferromagnetism, etc.) depends on or is determined by the behavior of its electrons. When describing the spin direction of a single electron, it is referred to as the magnetic dipole moment, while a magnetic domain refers to a bulk region of collective spins or uniformly aligned spins.
By default, it may be better to think of values in terms of ratios or slopes.	A guiding principle in analyzing systems or problems: scaling relationships, slopes, and derivatives often reveal more insight than absolute numbers.
Possible alternative name for microwave photon detector	Microwave photon radiation detector.
Everything is a transmission line, a capacitor (including self-capacitance), & inductor.	A broad conceptual notion in electronics and physics, emphasizing that all structures can be modeled as having transmission line properties, inherent capacitance, and inherent inductance.

Description of Spins & Spinors:


Electron spins, including all spin 1/2 particles, are physical realizations, out of many, of the abstract, mathematical spinor representations in nature. Interestingly, this is one example where an abstract mathematical object has experimentally measurable effects or direct experimental consequences. Spins in the technical sense generally refer to the description of an intrinsic angular momentum, meaning that it is purely quantum mechanical. In mathematics and physics, a spinor* is a type of object used to describe particles with half-integer spin (spin -1/2, spin +1/2, etc.). These objects transform in a particular way under rotations (technically under the group SU(2), which is the double cover of the rotation group SO(3)). Here, a 2π rotation changes the phase of a spinor by −1, meaning it does not return to its original state but instead acquires a sign flip. In quantum theory, classifying all possible particle types comes down to looking at irreducible representations of the Poincaré group (in special relativity) or the Galilean group (in non relativistic mechanics). Spin 1/2 emerges naturally when you look at certain irreducible representations-namely, those described by spinors. Any spin- fermion-such as quarks, protons (composite, but effectively spin 1/2 in total), and neutrinos-can also be described by spinors or "spinor formalism". Additional note: the spin of protons arises from a complex interplay of quark spins, gluon angular momentum, and orbital motion. Its total spin behaves like a fundamental fermion, but its substructure is different from an elementary particle.
*Spinors are two-component objects, they do return to the same quantum state after a 4π rotation but not after 2π.
Key terms: irreducible representations, experimental consequences

How to Generate a Pure Electric Field:

Method	Mechanism	Key Considerations	Applications
Photoelectric Effect	Light ejects electrons from a surface, creating charge imbalance	Requires photon energy > work function; works best in vacuum	Spacecraft charging, photoemission devices
Controlled Charge Separation	Electrons are emitted and collected on a secondary surface via photoemission, field emission, or thermionic emission	Requires an electron collector; prevents neutralization	Photoemission-based capacitors, charge storage
Vacuum Conditions	Electrons travel freely, leading to sustained electric fields	No surrounding medium to neutralize charge	Electron beam devices, vacuum tube applications
Voltage Bias Application	A potential difference guides photoelectrons to a specific region	Ensures continuous charge separation	Controlled electron beams, energy harvesting
UV Radiation-Induced Charging	High-energy UV photons eject electrons via the photoelectric effect	Effective in space or high-intensity UV environments	Spacecraft charging, UV-sensitive detectors
Solar Wind Charging	Plasma interactions induce surface charging and electron displacement	Occurs naturally in space; depends on plasma density and material properties	Spacecraft potential buildup, lunar dust levitation
Photoemission Cathodes	Use of a light-activated electron source in a circuit	Requires efficient cathode material	Photocathodes for electron guns, free-electron lasers

Cases of Quantized Light:

Category	Description	Examples
Photons (Fundamental Quanta)	Discrete energy packets of the electromagnetic field; primary quanta of light in quantum electrodynamics (QED) and quantum optics.	Spontaneous emission, stimulated emission, blackbody radiation, Compton scattering.
Quantized Modes of the Electromagnetic Field	The electromagnetic field can be described in terms of quantized oscillatory modes, even in the absence of real photons.	Vacuum fluctuations, zero-point energy, cavity QED, waveguide modes.
Coherent & Squeezed States	Special quantum states where light retains some classical properties but still exhibits quantum behavior.	Laser light (coherent states), squeezed vacuum states in quantum optics.
Polaritons (Hybrid Quasiparticles)	Light interacts with material excitations, forming mixed light-matter quasiparticles.	Exciton-polaritons (light + excitons in semiconductors), phonon-polaritons (light + lattice vibrations).
Plasmons & Surface Plasmon Polaritons (SPPs)	Quantized collective oscillations of free electrons in a metal, coupled with the electromagnetic field.	Surface-enhanced Raman spectroscopy (SERS), plasmonic waveguides, localized surface plasmons in nanoparticles.

Inspired by: Ezratty, Understanding Quantum Technologies, 2111.15352 (2024)
https://doi.org/10.48550/arXiv.2111.15352
https://creativecommons.org/licenses/by-nc-nd/4.0/

Definition of Arbitrary & Arbitrary Units:

Term	Definition
Arbitrary	Refers to chosen values or units that maintain internal consistency without relying on an external, standardized reference. Example: arbitrary units used in graphs and charts.
Arbitrary Units (a.u.)	Used in graphs and charts, they represent a consistent measure but do not correspond to a standardized physical unit. They are meaningful within the given context but are not directly comparable to a universal scale.

Key Differences of How Analytical Solutions are Defined & Used (Deterministic vs. Indeterministic):

Deterministic Analytical Solutions	Indeterministic Analytical Solutions
Definition: A solution that, given the same initial conditions, always produces the same result. These solutions are fully predictable and can be expressed in a closed mathematical form.	Definition: A solution where the outcome is probabilistic or dependent on unknown/uncontrollable factors, even if the underlying equations are well-defined.
Characteristics: - No randomness or probability involved. - Given a set of initial conditions and equations, the result is always the same. - Typically derived using exact algebraic, calculus-based, or differential equation methods.	Characteristics: - Involves probabilities or randomness in the results. - Repeating the same conditions does not always yield the same outcome. - Often appears in quantum mechanics, chaotic systems, and stochastic processes.
Examples: 1. Newton’s Laws of Motion: - If you apply a known force to an object, its acceleration and trajectory can be determined exactly. - Example: $( x(t) = x_0 + v_0t + \frac{1}{2}at^2 )$ (kinematics equation). 2. Ohm’s Law in Circuits: - $( V = IR )$ gives the exact voltage given current $( I )$ and resistance $( R )$. 3. Schrödinger's Equation for Simple Systems: - The time-independent Schrödinger equation for a hydrogen atom yields exact energy eigenvalues for electron states.	Examples: 1. Quantum Mechanics (Wavefunction Collapse): - The Schrödinger equation deterministically evolves a wavefunction, but upon measurement, the outcome is probabilistic. - Example: Measuring the spin of an electron in a superposition state gives a random outcome (e.g., 50% spin-up, 50% spin-down). 2. Radioactive Decay: - The decay of a single nucleus follows a probability distribution, not a deterministic function. - We can only predict half-life, but not when a specific atom will decay. 3. Chaotic Systems (Butterfly Effect): - Some classical systems, like weather models, follow deterministic equations but are highly sensitive to initial conditions, making long-term predictions effectively non-deterministic. 4. Monte Carlo Simulations: - Used in optimization and physics, these rely on random sampling to approximate solutions to complex problems.
Summary: - Deterministic solutions provide exact answers every time for given conditions.	Summary: - Indeterministic (or non-deterministic) solutions involve probabilities or sensitive dependencies, making exact results uncertain, even if the equations governing the system are known.

Inspired by: Gisin, Indeterminism in Physics and Intuitionistic Mathematics, Synthese 199, 13345–13371 (2021)
https://doi.org/10.1007/s11229-021-03378-z
http://creativecommons.org/licenses/by/4.0/

A Variety of Problem-Solving Approaches & Where They Tend to Fall Along the Two Dimensions:

Approximate --> Exact (top to bottom)
Intuitionistic --> Analytical (left to right)

	More Intuitionistic	Mixed/Hybrid	More Analytical
Highly Approximate (Top)	Pure Intuition - Gut feeling - Instinctive reactions	Trial-and-Error/ Rough Guess - Ad-hoc tinkering - Quick “guess-and-check” attempts	Back-of-the-Envelope Computations - Rapid mental math or simplified analysis - “Quick and dirty” calculations
Moderately Approximate	Educated Guesses/ Analogy - Intuition guided by analogies - Domain-specific “rules of thumb”	Heuristics - General problem-solving rules of thumb Metaheuristics - Genetic algorithms, evolutionary methods, etc.	Simplified Modeling/ Monte Carlo - Partial modeling or assumptions - Stochastic approximations and simulations
Moderately Exact	Intuitive Domain Expertise - Structured “gut sense” from years of practice	Systematic Heuristics - Deliberately applied heuristic sets Hybrid Methods - Combining data + experience for iteration	Algorithmic/ Structured - Well-defined step-by-step procedures - Many optimization or search algorithms
Highly Exact (Bottom)	(Rare purely “intuitive” exactness-often specialized or self-correcting)	Formalized Hybrid Methods - Constraint programming with heuristic guidance - High-level frameworks that incorporate both data and domain insight	Formal/ Deductive - Mathematical proofs - Exhaustive search - Rigorous deductive logic/ derivations

Categories of Mathematical Spaces

(Of which many carry direct physical interpretations)

Category	Examples	Origin	Usage
Pure mathematical spaces	Hilbert, Banach, Sobolev, manifold	Math only	Foundations, proofs, structures
Mathematical spaces used in physics	Phase space, Fock space, Fourier, Minkowski spacetime	Math → adopted by physics	Quantum mechanics, optics, relativity
Physical spaces expressed mathematically	Configuration space, Quantum Fourier Transform field, spacetime manifolds (Minkowski, de Sitter, anti-de Sitter)	Physics → formalized by math	Physical interpretation layers

Space Type	Core Structure	Physical Role
Euclidean space ℝⁿ	Flat metric space with positive-definite inner product	Classical geometry, nonrelativistic configurations
Hilbert space	Inner-product, complete	Quantum states, unitaries, measurements
Phase space	Symplectic manifold (x,p)	Classical & quasi-quantum dynamics
Fourier space	Dual frequency/momentum domain	Spectra, band structure, transforms
Configuration space	Position-domain manifold	Many-body kinematics
Momentum space	Conjugate to real-space	Scattering, dispersion
Fock space	Ladder-operator basis	Bosonic/fermionic modes
Density-matrix space	PSD trace-1 operators	Mixed states, noise, decoherence
Liouville space	Vectorized operators	Lindblad evolution, channels
Tensor-product space	Composite quantum system	Multiqubit states and correlations
k-space/Reciprocal space	Momentum lattice	Crystalline solids, phonons, electrons
Path-integral space	Functional trajectories	Quantum field theory
Minkowski spacetime	4D pseudo-Riemannian manifold with metric signature (−,+,+,+)	Special relativity, flat-space quantum field theory

Mathematical & Physical Spaces
├─ 1. Linear-Structure Spaces
│   ├─ 1.1 Hilbert Space (ℋ)
│   │   ├─ Complete inner-product space, norm from ⟨ψ|ψ⟩
│   │   ├─ Quantum states, operators, spectra
│   │   └─ Basis types: orthonormal, continuous, generalized (Dirac)
│   ├─ 1.2 Banach Space
│   │   ├─ Complete normed vector space
│   │   └─ Includes Lᵖ spaces (p ≥ 1)
│   ├─ 1.3 Function Spaces
│   │   ├─ L² space (square-integrable functions)
│   │   ├─ Sobolev spaces (differentiability + integrability)
│   │   └─ Schwartz space (rapidly decaying smooth functions)
│   └─ 1.4 Operator Spaces
│       ├─ Bounded operators B(ℋ)
│       ├─ Trace-class & Hilbert-Schmidt operators
│       └─ C*-algebras, von Neumann algebras
│
├─ 2. Dual & Transform Spaces
│   ├─ 2.1 Fourier Space/ Momentum Space
│   │   ├─ Dual to position representation
│   │   └─ Contains p-space wavefunctions ψ(p)
│   ├─ 2.2 Laplace Space/ s-Domain
│   │   ├─ Control theory, electronics, stability analysis
│   │   └─ Poles, residues, Green’s function structure
│   ├─ 2.3 Reciprocal Lattice Space (k-Space)
│   │   ├─ Crystals, band structures
│   │   └─ Brillouin zones, dispersion relations
│   └─ 2.4 Wavelet Space
│       ├─ Multiresolution analysis
│       └─ Time-frequency decomposition
│
├─ 3. Geometric & Symplectic Spaces
│   ├─ 3.1 Phase Space (x,p)
│   │   ├─ Classical: symplectic manifold with ω = dx∧dp
│   │   ├─ Quantum-quasi: Wigner function on phase space
│   │   └─ Used in quantum optics, tomography
│   ├─ 3.2 Configuration Space
│   │   ├─ Position-space domain of a system
│   │   └─ Many-body: ℝ³N
│   ├─ 3.3 Momentum Space (p-space)
│   │   └─ Canonically conjugate to real-space
│   ├─ 3.4 Symplectic Vector Spaces
│   │   └─ Sp(2n) transformations (Gaussian optics, bosonic modes)
│   └─ 3.5 Riemannian & Pseudo-Riemannian Manifolds
│       ├─ Minkowski spacetime (flat special-relativistic space-time)
│       ├─ Curved space-time (general relativity)
│       └─ Quantum fields in curved backgrounds
│
├─ 4. Statistical, Information, & Probability Spaces
│   ├─ 4.1 Probability Space (Ω, F, P)
│   ├─ 4.2 Statistical Manifolds
│   │   ├─ Fisher information geometry
│   │   └─ Cramer-Rao bounds, QFI links
│   ├─ 4.3 Density-Matrix State Space
│   │   ├─ Positive semidefinite, trace-1 operators
│   │   ├─ Mixed states, purifications
│   │   └─ Bloch sphere (qubit subset)
│   └─ 4.4 Liouville Space (Superoperator space)
│       ├─ Vectorized density matrices |ρ⟩⟩
│       └─ Lindbladian dynamics, quantum channels
│
├─ 5. Computational & Algorithmic Spaces
│   ├─ 5.1 Tensor Product Spaces
│   │   ├─ Many-qubit systems
│   │   └─ Tensor network embeddings
│   ├─ 5.2 State-Vector Simulation Space
│   │   └─ 2ⁿ-dimensional complex vectors
│   ├─ 5.3 Circuit Space
│   │   ├─ Sequence space of unitary gates
│   │   └─ Error channels as CPTP maps
│   └─ 5.4 Feature Spaces (Machine Learning)
│       ├─ Kernel Hilbert space (RKHS)
│       └─ Quantum kernels, QML embeddings
│
├─ 6. Algebraic & Representation Spaces
│   ├─ 6.1 Lie Group Manifolds
│   │   ├─ SU(2), SU(3), U(1), SO(n)
│   │   └─ Quantum gates, rotations
│   ├─ 6.2 Representation Spaces
│   │   ├─ Irreducible representations (irreps)
│   │   ├─ Angular momentum, Clebsch-Gordan
│   │   └─ Poincaré group representations on Minkowski spacetime
│   ├─ 6.3 Projective Hilbert Space
│   │   └─ Physical states modulo global phase
│   └─ 6.4 Fock Space
│       ├─ Bosons, photons, phonons, magnons
│       └─ Creation/annihilation ladder structure
│
└─ 7. Topological, Categorical, & Quantum-Field Spaces
    ├─ 7.1 Topological Spaces
    │   ├─ Open sets, continuity, homotopy
    │   └─ Topological invariants (Chern numbers)
    ├─ 7.2 Fiber Bundles
    │   ├─ Gauge fields, Berry curvature
    │   └─ Connections, holonomy
    ├─ 7.3 Path-Integral Configuration Space
    │   ├─ Sum over histories
    │   └─ Field trajectories as points in function space
    ├─ 7.4 Quantum Field Configuration Space
    │   ├─ Fields φ(x) on Minkowski or curved space-time as infinite-dimensional points
    │   └─ Renormalization flow spaces
    └─ 7.5 Category-Theoretic State Spaces
        ├─ Monoidal categories
        ├─ Quantum circuits as morphisms
        └─ Topological quantum computation (anyons)

Acronyms

SU(n) = Special unitary group of degree n

SO(n) = Special orthogonal group of degree n

U(1) = Unitary group of degree 1 (phase rotations)

How the Categorized Mathematical Spaces Connect

Hilbert Space
├─ connects to Fourier Space via unitary transforms
├─ connects to Phase Space via Wigner transform
├─ connects to Density-Matrix Space via |ψ⟩⟨ψ|
├─ connects to Tensor Spaces via ℋ⊗ℋ
└─ connects to Fock Space through occupation-number basis

Phase Space
├─ connects to Symplectic Spaces through canonical transformations
├─ connects to Fourier Space through characteristic functions
└─ connects to Quantum Optics via Wigner, Husimi-Q, Glauber-S

Fourier Space
├─ connects to k-Space in crystals
├─ connects to Momentum Space via p = ħk
└─ connects to Control/Signal domains through Laplace and z-transforms

Density-Matrix Space
├─ connects to Liouville Space via vectorization
├─ connects to Projective Space (pure-state boundary)
└─ connects to Statistical Manifolds via quantum Fisher information

Fock Space
├─ connects to Hilbert Space through mode decomposition
├─ connects to Path Integrals via coherent state representations
└─ connects to Topological Spaces in anyonic field theories

Lie Group/Representation Spaces
├─ connects to Hilbert Space via unitary reps
├─ connects to Quantum Circuits via SU(2ⁿ)
└─ connects to Geometry through SU(2)/SO(3) isomorphisms

Minkowski Spacetime / Space-Time Manifolds
├─ connects to Quantum Field Configuration Space as the flat background manifold
├─ connects to Hilbert Space via relativistic quantum state spaces
├─ connects to Lie Group/Representation Spaces through Poincaré group symmetries
└─ connects to Fiber Bundles via gauge fields defined over space-time

Basic Math Symbols ≠ ± ∓ ÷ × ∙ – √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °

Geometry Symbols ∠ ∟ ° ≅ ~ ‖ ⟂ ⫛

Algebra Symbols ≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘∏ ∐ ∑ ⋀ ⋁ ⋂ ⋃ ⨀ ⨁ ⨂ 𝖕 𝖖 𝖗 | 〉

Set Theory Symbols ∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟

Logic Symbols ¬ ∨ ∧ ⊕ --> <-- ⇒ ⇐ ↔ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣

Calculus & Analysis Symbols ∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ

Greek Letters 𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔

Example Mathematical Expressions Along with Their Descriptions

Symbol	Name/ phrase	Meaning
$(a_n)$	sequence $a_n$	An infinite list of real numbers: $a_1, a_2, a_3, \dots$
$a \in \mathbb{R}$	$a$ is in $\mathbb{R}$	$a$ is a real number
$\mathbb{R}$	real numbers	The set of all real numbers
$\mathbb{N}$	natural numbers	${1, 2, 3, 4, \dots}$ (positive integers)
$n$	index	A natural-number index $1, 2, 3, \dots$
$\lim_{n\to\infty} a_n$	limit of $a_n$ as $n\to\infty$	The value $a_n$ approaches as $n$ gets arbitrarily large
$(a_n) \to a$	“ $a_n$ tends to $a$ ”	Another notation for $\lim_{n\to\infty} a_n = a$
$\forall$	“for all” / “for every”	Universal quantifier
$\exists$	“there exists”	Existential quantifier
$\epsilon$	epsilon	A positive real number that represents an error tolerance
$\epsilon > 0$	epsilon is positive	We only consider strictly positive tolerances
$N$	capital $N$	A natural number beyond which something nice happens in the sequence
$n \ge N$	$n$ greater than or equal to $N$	“$n$ is at least $N$”
$\implies$	implies	Logical implication: if left side holds, then right side must hold
$\lvert x \lvert$	absolute value of $x$	Distance from $x$ to $0$ on the real line; $\lvert x \lvert \ge 0$
$\lvert a_n - a\lvert < \epsilon$	distance condition	The term $a_n$ is within $\epsilon$ units of the limit $a$

Hyperoperations & Tetrations

Hyper-operations
├─ Level 1: Addition
│   ├─ Visible: a + 0
│   └─ Invisible "add 0": a
├─ Level 2: Multiplication
│   ├─ Visible: a · 1
│   └─ Invisible "×1": a
├─ Level 3: Exponentiation
│   ├─ Visible: a^1
│   └─ Invisible exponent: a
└─ Level 4: Tetration
    ├─ Visible: {}^1 a   (height 1)
    ├─ Visible: {}^2 a   (a^a)
    └─ Invisible tetration:
        • strict identity choice: a  = {}^1 a
        • custom default tower: Tet(a) := {}^2 a = a^a

Level	Operation	Full form	Default parameter made invisible	Invisible form
1	Addition	( $a+b$ )	Additive identity (0)	( $a=a+0$ )
2	Multiplication	( $a \cdot b$ )	Multiplicative identity (1)	( $a=a \cdot 1$ )
3	Exponentiation	( $a^b$ )	Exponent (1)	( $a=a^1$ )
4	Tetration	$\left({ }^n a=a \uparrow \uparrow n\right)$	Height (1) (identity)	$\left(a={ }^1 a\right)$

Below is an Example by (O.J.B.) on How to Substitute with Custom Variables:

Note: the W, x, y, z, T variables in the example above are merely substites & do not correspond to any physical variables.

All The Physics You Need, Curated by (O.J.B.):

Topic	Source
How to Succeed at Physics Without Really Trying	Physics with Elliot
The Most Important Math Formula for Understanding Physics	Physics with Elliot
The Single Basic Concept Found in (Almost) All Fundamental Physics Equations	Parth G
To Master Physics, First Master the Harmonic Oscillator	Physics with Elliot
To Master Physics, First Master the Rotating Coordinate System	Dialect
5 Methods for Differential Equations	Physics with Elliot
Imaginary Numbers	Parth G
Why Momentum in Quantum Physics is Complex	Parth G
The Wave Equation	Parth G
Solving the Wave Equation	Parth G
Poisson's Equation for Beginners	Parth G
Eigenvalue Equation	Parth G
Solving the Schrödinger Equation	Parth G
How Energy is Created Quantum Mechanics [Creation & Anihilation Operators]	Parth G
Quantum Operators & Commutators	Parth G
Quantum Physics Becomes Intuitive with this Theorem, Ehrenfest's Theorem EXPLAINED	Parth G
What Is Ehrenfest Theorem, Ehrenfest Theorem Explained, Ehrenfest Theorem Quantum Mechanics	Physics for Students
Perturbation Theory	Parth G
Matrices	Parth G
Understand Tensors Like a Physicist (The Easy Way)	Physics - Problems & Solutions
Lagrangian & Hamiltonian Mechanics	Physics with Elliot
The Kronecker Delta	Alexander Fufaev
The Levi-Civita Symbol & Kronecker Delta	Jeffrey Chasnov
Kronecker Delta & Levi-Civita Symbol	Jeffery Chasnov
Maxwell's Equations Explained	Parth G
Animated Physics Lectures	ZAP Physics
More Animated Physics Lectures	Alexander Fufaev
Even More Animated Physics Lectures	Dr. Elliot Schneider
Physical Sciences & Engineering	Dr. Jordan Edmunds
Maths of Quantum Mechanics Playlist	Quantum Sense
Quantum Harmonic Oscillators	Pretty Much Physics
Dirac Equation Playlist	Pretty Much Physics
Approximation Methods	TMP Chem
Quantum Information Science Playlists	Prof. Artur Ekert
Griffiths Quantum Mechanics Playlist	Nick Heumann
Modern Physics	Professor Dave Explains
Quantum Physics I	MIT OCW
Quantum Physics II	MIT OCW
Quantum Physics III	MIT OCW
Physical Chemistry	MIT OCW
Physical Chemistry	Prof. Derricotte
Physical Chemistry I	Stuart Winikoff
Physical Chemistry II	Stuart Winikoff
Quantum Chemistry	Trent M. Parker
Quantum Transport	Prof. Sergey Frolov
Quantum Many-Body Physics	Prof. Luis Gregório Dias
Quantum Matter	Prof. Steven Simon
Quantum Optics	Prof. Carlos Navarrete-Benlloch
Quantum Optics	Prof. Immanuel Bloch
Topological Quantum Matter	Weizmann Institute of Science
Quantum Field Theory Playlist	Nick Heumann
Quantum Field Theory Playlist	Dietterich Labs
Relativistic Quantum Field Theory Playlist	MIT OCW
Important Notes & Physics Etiquettes	Physics with Elliot
Math Notes for Quantum Information Science	Introduction to Quantum Information Science
Time-Dependent Quantum Mechanics & Spectroscopy Notes	UChicago
Solid-State Physics	Prof. M. S. Dresselhaus
Transport in Semiconductor Mesoscopic Devices	David K. Ferry (Book 1) / David K. Ferry (Book 2)

Additional Physics & Mathematics Resources:

Topic	Source
Introduction to Mathematical Reasoning Playlist	Knop's Course
General Mathematical Playlists	Faculty of Khan
Mathematical Physics Playlists	Dietterich Labs
Physics Playlists	Physics for Students
Physics Education Playlists	Acephysics
Geometric Algebra - Why	Parker Glynn-Adey
Geometric Algebra - Why 2	Bivector
Zero to Geo[metric Algebra]	sudgylacmoe
Differential Geometric Algebra	Crucial Flow Research
Advanced Mathematics	The Bright Side of Mathematics
More Advanced Mathematics	Cofiber
Lessons on Prerequisitcs for Quantum & Related Topcs	XylyXylyX
Spinors Playlist	eigenchris
Weinberg's Lectures on Quantum Mechanics Playlist	Physics Daemon
Thermodynamics & Statistical Physics Playlist	Pazzy Boardman
Statistical Mechanics & Thermodynamics Playlist	Physics Daemon
Solid State Devices Playlist	nanohubtechtalks
QuTech360 Seminars	QuTech Academy
Quantum Playlists	Nick Heumann University
STEM Full Course Playlists	Academic Lesson

Shortcuts in Mathematics and Physics:


Numberphile, Interview-Style Playlists
Sixty Symbols, Interview-Style Playlists
UoN Physics, Interview-Style Playlists
DiBeos, Interview-Style Playlists
Amazing Things You Can Do in Geometric Algebra
Related Rates - Conical Tank, Ladder Angle & Shadow Problem, Circle & Sphere - Calculus
Understand Calculus in 35 Minutes
Chain Rule With Partial Derivatives - Multivariable Calculus
Integration Using The Substitution Rule
Partial Derivatives - Multivariable Calculus
Vector Fields, Divergence, and Curl
Mathematical Foundations of Quantum Mechanics
Mastering Quantum Mechanics Through Problems

To "Do Physics"

When people say “what’s the physics?” they are very often asking, first, for stimulus-response behavior and the shapes of the resulting curves. However, to be more complete, “the physics” also includes the precise words and symbols we use, the governing laws and constraints, the parameters and units, the valid regimes and approximations, the noise and uncertainty, and the geometry and boundary conditions that make those curves look the way they do.

In most scientific and engineering conversations, “what’s the physics?” quickly centers on how outputs change with inputs and what the curves look like. However, the full answer also, and necessarily, includes the terminology, parameters, laws and constraints, scales and geometry, and the noise and uncertainty that make those curves meaningful, portable, and predictive.

When someone asks for “the physics,” they are asking, in plain terms, “If I poke this thing in different ways, how does it react, and why?” You describe:

The inputs you can change, like pushing, heating, applying voltage, shining light, changing shape.
The outputs that change, like stretch, temperature, current, brightness, frequency.
The graph shapes: straight lines (linear), bowed curves (non-linear), loops (hysteresis), steps (thresholds), peaks (resonances).
The names and numbers that summarize those graphs: slope, spring constant, resistance, conductivity, susceptibility, time constant, quality factor.
The simple rules that tie inputs to outputs: Hooke’s law, Ohm’s law, Beer–Lambert law, the ideal-gas law, and so on.
The conditions that matter: size, temperature, environment, and the way the thing is built.

“What’s the physics?” usually means assembling eight interlocking pieces:

Here is the list rewritten using a Roman numeral hierarchy:

I. Phenomenology (measurable behavior).

Empirical relations and response surfaces $y=f(x;\theta)$ that map stimuli to responses, with uncertainty bands. This includes constitutive relations (e.g., stress–strain), transfer functions in the frequency domain, and susceptibilities $\chi(\omega)$.

II. Mechanism (micro to macro).

The microscopic or mesoscopic model-Hamiltonians, free energies, rate equations, transport equations-from which the phenomenology can be derived or approximated. Examples: Kubo linear-response for $\chi(\omega)$, Landau theory near phase transitions, Boltzmann transport for conductivity.

III. Constraints (principles).

Conservation of energy, momentum, charge; causality; passivity; stability; reciprocity or non-reciprocity; and symmetry requirements (Noether-style thinking). These prune which curves are even allowed.

IV. Scales and nondimensional groups.

Which length, time, and energy scales dominate; which Reynolds, Peclet, Deborah, or quality-factor $Q$ numbers control the regime; what asymptotics apply (low-frequency, high-field, long-time, dilute-limit).

V. Geometry and boundary conditions.

Shape, topology, interfaces, terminations, fixtures, packaging. Many “curve shapes” are really boundary-condition effects.

VI. Noise and fluctuations (and information).

Variances, spectral densities $S(\omega)$, fluctuation–dissipation links, and the signal chain that makes responses observable with finite signal-to-noise ratio.

VII. Terminology and parameters.

The controlled vocabulary and symbols, the units and dimensional analysis, and the calibrated parameters with error bars. This is absolutely part of “the physics,” because unambiguous language and parameterization make the model testable and transferable.

VIII. Validation and extrapolation.

Design-of-experiments, cross-validation across stimuli and geometries, and the limits where the model fails or needs higher-order terms.

“What’s the physics?”
├─ Phenomenology --> response curves, transfer functions, susceptibilities
│   └─ Parameters --> slopes/gains, thresholds, time constants, Q, χ(ω)
├─ Mechanism --> Hamiltonian/ free-energy/ transport picture
│   └─ Derivations --> linear response, perturbation, asymptotics
├─ Constraints --> conservation, causality, stability, symmetry
├─ Scales --> dominant lengths/times/energies; dimensionless groups
├─ Geometry/BCs --> shape, interfaces, packaging, loading
├─ Noise/Uncertainty --> PSDs, SNR, confidence intervals
├─ Terminology --> standardized variables, units, regimes
└─ Validation --> DOEs, cross-checks, regime maps, failure modes

When does a stimulus-response study count as physics?

Activity	Counts as physics when…	Why it qualifies	Typical deliverables	Where it sits
Curves-only catalog	…the curves are tied to explicit hypotheses and are reproducible with an uncertainty model.	Falsifiable claims, replicable procedures, quantified uncertainty.	Raw data, calibration notes, uncertainty budget, curve fits with confidence intervals.	Experimental physics, metrology.
Parameter extraction	…parameters map to physical quantities and predict new data without refitting.	From description to mechanism-anchored prediction.	(Q), thresholds, rate constants, susceptibilities ( $\chi(\omega)$ ) with error bars.	Phenomenology, transport, materials.
Scaling/Universality	…dimensionless groups collapse many geometries and conditions.	Reveals law-like structure beyond one setup.	Master curves, regime maps, similarity rules.	Statistical/condensed-matter physics.
Mechanistic modeling	…a Hamiltonian, free energy, or transport model reproduces the response and passes cross-checks.	Connects micro to macro, enables design.	Linear/nonlinear response, dispersion relations, predictions at new stimuli.	Theory ↔ experiment bridge.
DOE-driven exploration	…factorial sweeps and blocking separate main effects from interactions.	Efficient, defensible inference.	Factor effects, interaction plots, power analysis.	Experimental design & analysis.
Reporting & openness	…procedures, data, and code permit reproduction.	Objectivity, community testing.	Reproducible notebooks, data + metadata.	Best practice across physics.

So, if you provide all of the analyzed data for these responses for a system of interest or set of physical materials, then one could present it as a thesis?

To be a thesis, you do not merely show what happens when you poke the system. You also, and importantly, explain why those curves look that way, how to predict them, how certain you are, and what is new that the field did not have before. So, yes: a thorough response (atlas can be a thesis) if it either discovers something new, or builds a reusable, tested model, or creates a widely useful method and dataset that others can build on.

Stimulus–Response Atlas --> “Physics Thesis”
├─ Research Question & Novelty
│   ├─ New mechanism?     ├─ New method?     └─ New scaling law?
├─ Experimental Design (DOE)
│   ├─ Factors/levels  ├─ Replication  ├─ Randomization  └─ Blocking
├─ Measurement & Uncertainty
│   ├─ Calibration tree  ├─ GUM model  └─ Sensitivity & Monte Carlo
├─ Response Modeling
│   ├─ Linear response (Kubo/susceptibility) --> KK causality check
│   └─ Nonlinear extensions, saturation, hysteresis, bifurcations
├─ Geometry & Boundary Conditions
│   ├─ Shape/topology  └─ Interfaces, packaging, loading
├─ Scaling & Regimes
│   ├─ Dimensionless groups  ├─ Curve-collapse  └─ Regime map
├─ Validation
│   ├─ Hold-out predictions  └─ Cross-geometry/temperature tests
└─ Reuse & Community
    ├─ FAIR data+code  ├─ Data Availability Statement  └─ Reproducible notebooks

A practical scaffold for a data-driven physics thesis

Chapter 1 - Problem, contributions, claims. State the question, the gap, and exactly what is original. Anchor against the “original contribution” criterion.
Chapter 2 - Methods, calibration, DOE. Define stimuli, responses, states, controlled variables, and geometry. Lay out DOE, replication, and randomization. Document calibration chains and uncertainty budgets.
Chapter 3 - Response surfaces and parameter extraction. Present full maps $y=f(x_1,x_2,\dots)$, residuals, confidence bands, and parameter-identifiability diagnostics.
Chapter 4 - Mechanism and constraints. Derive linear- and, where needed, non-linear response. Show that susceptibilities satisfy causality and Kramers–Kronig; explain dispersion/loss trade-offs.
Chapter 5 - Scaling and universality. Build non-dimensional groups, attempt curve-collapse across geometries and temperatures, and articulate regime maps.
Chapter 6 - Validation and prediction. Predict unseen conditions and compare to new experiments; quantify predictive uncertainty.
Chapter 7 - Reuse package. Release data, code, and metadata with a clear Data Availability Statement and FAIR checklist; include a “parameter-extraction recipe.”

Practically and systematically, the data-driven physics thesis has four pillars:

Originality. Frame a crisp research question and add something the field did not have: a mechanism, a model, a scaling law, a method, or a high-quality, FAIR-ready dataset and codebase. (Universities state “original contribution” as a core requirement.)
Rigor in measurement and uncertainty. Report how well you know slopes, thresholds, time-constants, and susceptibilities, using accepted standards such as ISO GUM and NIST TN 1297 to propagate and present uncertainties.
Design of experiments (DOE). Use DOE to cover factor space efficiently and defensibly, then model main effects and interactions so your conclusions generalize.
Stewardship and reuse. Make your data Findable, Accessible, Interoperable, and Reusable (FAIR), and include a Data Availability Statement aligned with current journal policies (APS and others).

Is this 'doing physics'? How does it fall under such a category?

Mapping stimuli to responses, with explicit uncertainty, mechanisms, and predictions, is exactly “doing physics.” It lives in experimental physics and phenomenology, it is scaffolded by metrology and DOE, and it becomes deepest when your response curves are not just described, but explained, constrained, and successfully predicted across new conditions.

"Stimulus --> response" atlas falls squarely under experimental physics and phenomenology, supported by metrology and modeling:

Experimental physics & phenomenology: you establish quantitative laws linking inputs to outputs, often via susceptibilities, transfer functions, or constitutive relations, and you compare these to theory. This "bridge from measurements to models" is called phenomenology in physics.
Metrology & uncertainty: you treat every measured curve as an estimate with uncertainty, you define the measurand precisely, and you propagate uncertainty per recognized guides like the Guide to the Expression of Uncertainty in Measurement (GUM).
Design of experiments (DOE): you plan factor sweeps and replications so that your inferences are efficient and defensible, not accidental.
Mechanism & response theory: you connect the curves to a mechanism, often via linear-response theory (Kubo-style), and you use that structure to predict new conditions.
Scientific method: your claims are framed so that an observation could falsify them, therefore they are scientific claims, not irrefutable descriptions.

"Doing physics" is the conjunction of measuring, modeling, uncertainty-quantifying, and testing, with an eye toward prediction and explanation. Mapping stimuli to responses, with explicit uncertainty, mechanisms, and predictions, is exactly “doing physics.” It lives in experimental physics and phenomenology, it is scaffolded by metrology and DOE, and it becomes deepest when your response curves are not just described, but explained, constrained, and successfully predicted across new conditions.

Stimulus --> Response Atlas
├─ Phenomenology (measured laws)
│   ├─ Constitutive relations, transfer functions
│   └─ Parameter extraction with CIs
├─ Metrology (how sure are we?)
│   ├─ Measurand definition, calibration chain
│   └─ GUM-style uncertainty propagation
├─ Design of Experiments
│   ├─ Factors, levels, replication, blocking
│   └─ Interaction detection, efficiency
├─ Mechanism (why those curves?)
│   ├─ Linear-response (Kubo) & susceptibilities
│   └─ Nonlinear extensions, thresholds, hysteresis
├─ Prediction & Falsifiability
│   ├─ Out-of-sample tests, regime maps
│   └─ Clear potential falsifiers
└─ Communication & Reuse
    ├─ Reproducible code/data
    └─ Terminology, units, controlled vocabulary

Example “stimulus --> response” table structure

Stimuli (factors)	Responses (observables)	Derived parameters	Uncertainty budget entries	Notes on geometry/BCs
Amplitude, frequency, temperature, bias/field, geometry, environment	Gain, slope, time-constant, resonance peak ($\omega_0$), linewidth ($\Delta\omega$), hysteresis width, switching rate	( $Q=\omega_0/\Delta\omega$ ), ( $\chi(\omega)$ ), threshold ( $x_\mathrm{th}$ ), nonlinearity ($k_3$), diffusion ($D$)	Type A: repeatability, drift; Type B: calibration certificates, resolution, model truncation	Include fixtures, terminations, interfaces, packaging; state boundary conditions explicitly

Original contribution is a universal doctoral criterion across leading universities; relying solely on descriptive compilation is typically not sufficient without a novel, validated insight or resource.
Measurement uncertainty should follow recognized metrology standards (GUM, NIST TN 1297), which increases credibility and transferability.
DOE ensures your map of conditions is efficient and defensible, and it improves the power of any conclusions you generalize.
FAIR stewardship and Data Availability Statements are increasingly required or encouraged by major publishers and societies, including APS, making your thesis more citable and reusable.
Response-function formalism (e.g., Kubo, susceptibilities) and causality constraints (Kramers–Kronig) turn curves into physics by enforcing what shapes are physically allowed.

Table of Critical Equations to Know:

#	Equation name	Equation (LaTeX)	Category/ context	Key symbols/ notes
1	Classical harmonic oscillator	$\dfrac{d^2 x}{dt^2} + \omega_0^2 x = 0$	Second‑order linear ODE; mechanics, circuits, fields	Undamped motion with angular frequency $\omega_0$. Damping adds term $2\zeta\omega_0,dx/dt$.
2	Wave (d’Alembert) equation	$\dfrac{\partial^2 u}{\partial t^2} = v^2 \nabla^2 u$	Hyperbolic PDE; propagation of waves	$u$ displacement, field, or potential; $v$ wave speed. In electromagnetism, $v=c$ in vacuum.
3	Laplace’s equation	$\nabla^2 \phi = 0$	Source‑free limit of Poisson; potential theory	Describes steady‑state fields with no local sources (electrostatics in charge‑free regions, incompressible flow streamfunctions, gravitational potential in vacuum).
4	Heat (diffusion) equation	$\dfrac{\partial u}{\partial t} = \alpha \nabla^2 u$	Parabolic PDE; heat flow, diffusion, probability densities	$u$ temperature or concentration, $\alpha = k/(\rho c_p)$ thermal diffusivity.
5	Poisson equation	$\nabla^2 \phi(\mathbf{r}) = -\rho(\mathbf{r}) / \varepsilon_0$	Elliptic PDE; electrostatics, gravitation	$\phi$ potential, $\rho$ charge density, $\varepsilon_0$ vacuum permittivity. In inhomogeneous media: $\nabla\cdot(\varepsilon \nabla\phi) = -\rho$.
6	Fourier transform (one common convention)	$F(k) = \displaystyle\int_{-\infty}^{\infty} f(x),e^{-ikx},dx,\quad f(x) = \dfrac{1}{2\pi}\int_{-\infty}^{\infty} F(k),e^{ikx},dk$	Spectral decomposition; signals, PDEs, quantum mechanics	Other conventions move the $2\pi$ factors or change the exponent sign. Choice must be consistent between transform and inverse.
7	Power conversion efficiency	$\eta = P_{\mathrm{out}} / P_{\mathrm{in}}$	Figure of merit; energy and power devices	$\eta$ dimensionless efficiency; often expressed as a percentage. Can also be defined for energies, work, photons, etc.
8	Ohmic conductivity (Ohm’s law, differential form)	$\mathbf{J} = \boldsymbol{\sigma} \mathbf{E}$	Constitutive relation; transport, electronics	$\mathbf{J}$ current density, $\mathbf{E}$ electric field, $\boldsymbol{\sigma}$ conductivity (scalar or tensor, often frequency dependent).
9	Green’s function definition	$L,G(\mathbf{r},\mathbf{r}') = \delta(\mathbf{r}-\mathbf{r}')$	Impulse response of linear differential operator $L$	Once $G$ is known, solution of $L u = f$ with given boundaries is $u(\mathbf{r}) = \int G(\mathbf{r},\mathbf{r}') f(\mathbf{r}'),d\mathbf{r}'$ (plus homogeneous solution).
10	Navier–Stokes equation (incompressible Newtonian fluid)	$\rho!\left(\dfrac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u}\right) = -\nabla p + \mu \nabla^2\mathbf{u} + \mathbf{f},\quad \nabla\cdot\mathbf{u}=0$	Continuum momentum balance for viscous fluids	$\mathbf{u}$ velocity, $p$ pressure, $\rho$ density, $\mu$ dynamic viscosity, $\mathbf{f}$ body force density (e.g., gravity). Derived from conservation of momentum plus constitutive law for Newtonian stress.
11	Fick’s first law of diffusion	$\mathbf{J} = -D \nabla c$	Constitutive law; mass transport, random‑walk limit	$\mathbf{J}$ diffusive flux, $D$ diffusion coefficient, $c$ concentration. Minus sign gives flux from high to low $c$.
12	Fick’s second law (diffusion equation)	$\dfrac{\partial c}{\partial t} = D \nabla^2 c$	Parabolic PDE; diffusion, random walks, probability	For constant $D$. More generally: $\partial c/\partial t = \nabla\cdot(D\nabla c)$.
13	Mean free path	$\ell = 1 /(n \sigma_{\mathrm{sc}})$	Kinetic theory; transport in gases, solids	$\ell$ average distance between scattering events, $n$ number density of scatterers, $\sigma_{\mathrm{sc}}$ scattering cross section. For hard spheres in a gas, $\ell = 1/(\sqrt{2} n \sigma)$.
14	Maxwell’s equations (microscopic, SI, differential form)	$\nabla\cdot\mathbf{E} = \rho/\varepsilon_0,;\nabla\cdot\mathbf{B} = 0,;\nabla\times\mathbf{E} = -\dfrac{\partial\mathbf{B}}{\partial t},;\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\dfrac{\partial\mathbf{E}}{\partial t}$	Classical electromagnetism	$\mathbf{E}$ electric field, $\mathbf{B}$ magnetic flux density, $\rho$ charge density, $\mathbf{J}$ current density, $\varepsilon_0,\mu_0$ vacuum permittivity and permeability.
15	Continuity equation (conserved scalar)	$\dfrac{\partial \rho}{\partial t} + \nabla\cdot\mathbf{J} = 0$	Local conservation of charge, mass, probability, etc.	$\rho$ density of conserved quantity, $\mathbf{J}$ corresponding flux or current density. In EM, follows from Maxwell’s equations.
16	Resonant angular frequency	$\omega_0 = \sqrt{k/m}$ (mechanical), $;\omega_0 = 1/\sqrt{LC}$ (electrical)	Simple harmonic oscillator, LC resonator	$k$ spring constant, $m$ mass; $L$ inductance, $C$ capacitance. Gives natural oscillation frequency in radians per second.
17	Geometric (Clifford) algebra identities	$ab = a\cdot b + a\wedge b,\quad e_i e_j + e_j e_i = 2\delta_{ij}$	Unified algebra of vectors, bivectors, spinors	Geometric product decomposes into symmetric inner product and antisymmetric exterior product. Basis vectors $e_i$ generate the metric via $e_i^2 = 1$ (Euclidean) or $\pm 1$ (general signatures).
18	Helmholtz equation	$(\nabla^2 + k^2)\psi(\mathbf{r}) = 0$	Time‑harmonic wave fields	$k = \omega/c$ wavenumber. Appears after separation of variables in the wave equation for sinusoidal time dependence.
19	Poynting vector and Poynting theorem (local EM energy conservation)	$\mathbf{S} = \mathbf{E}\times\mathbf{H},\quad \dfrac{\partial u_{\mathrm{EM}}}{\partial t} + \nabla\cdot\mathbf{S} = -\mathbf{J}\cdot\mathbf{E}$	Electromagnetic energy flow and power dissipation	$\mathbf{S}$ energy flux density, $\mathbf{H}$ magnetic field strength, $u_{\mathrm{EM}}$ EM energy density. Right‑hand term is power delivered to charges per unit volume.
20	Quality factor of a resonance	$Q = \omega_0 / \Delta\omega$	Dimensionless resonance sharpness; oscillators, cavities, filters	$\Delta\omega$ full width at half maximum. Equivalent definition: $Q = 2\pi \times (\text{energy stored} / \text{energy lost per cycle})$.
21	Poisson–Boltzmann equation (dimensionless, symmetric electrolyte)	$\nabla^2 \psi = \kappa^2 \sinh \psi$	Nonlinear elliptic PDE; screened Coulomb potentials in electrolytes, plasmas, semiconductors	$\psi = ze\phi/(k_{\mathrm{B}}T)$ dimensionless potential, $\kappa^{-1}$ Debye length. For general ionic mixtures: $\nabla^2\phi = -\rho_f/\varepsilon - \dfrac{1}{\varepsilon}\sum_i z_i e n_{i0} e^{-z_i e\phi/(k_\mathrm{B}T)}$.
22	Bragg’s law	$n\lambda = 2 d \sin\theta$	Wave interference; x‑ray, neutron, electron diffraction	$n$ integer order, $\lambda$ wavelength, $d$ lattice plane spacing, $\theta$ glancing angle between beam and planes.
23	Classical anharmonic oscillator (cubic example)	$\dfrac{d^2 x}{dt^2} + \omega_0^2 x + \alpha x^3 = 0$	Nonlinear oscillator; perturbation theory, nonlinear dynamics	$\alpha$ controls strength of nonlinearity. Corresponds to potential $V(x) = \tfrac{1}{2}m\omega_0^2 x^2 + \tfrac{1}{4}\alpha x^4$.
24	Shot‑noise current (RMS in bandwidth $\Delta f$)	$\Delta I_{\mathrm{rms}} = \sqrt{2 q I,\Delta f}$	Electronic/ photon counting noise; Poisson statistics	$I$ average current, $q$ elementary charge, $\Delta f$ measurement bandwidth. Current spectral density $S_I = 2 q I$.
25	Time‑dependent Schrödinger equation	$i\hbar,\dfrac{\partial \psi(\mathbf{r},t)}{\partial t} = \hat{H},\psi(\mathbf{r},t)$	Fundamental quantum evolution law	$\hat{H}$ Hamiltonian operator (kinetic + potential). For one nonrelativistic particle: $\hat{H} = -\hbar^2\nabla^2/(2m) + V(\mathbf{r},t)$.
26	Quantum harmonic oscillator (1D)	$\Big[-\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + \tfrac{1}{2}m\omega_0^2 x^2\Big]\psi_n(x) = E_n \psi_n(x)$, $E_n = \hbar\omega_0 (n + \tfrac{1}{2})$	Exactly solvable model; quantized vibrations, modes	$n=0,1,2,\dots$; ladder operators connect eigenstates. Central in quantization of fields and cavity modes.
27	Time‑independent Schrödinger equation (TISE)	$\Big[-\dfrac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\Big]\psi(\mathbf{r}) = E,\psi(\mathbf{r})$	Stationary states; eigenvalue problem for $\hat{H}$	Separation of variables with $\psi(\mathbf{r},t) = \psi(\mathbf{r})e^{-iEt/\hbar}$ reduces the TDSE to this eigenproblem. Discrete $E_n$ give bound states, continuous $E$ scattering states.
28	Ehrenfest’s theorem (general operator form)	$\dfrac{d}{dt}\langle \hat{A} \rangle = \dfrac{1}{i\hbar},\langle[\hat{A},\hat{H}]\rangle + \left\langle \dfrac{\partial \hat{A}}{\partial t}\right\rangle$	Bridge between quantum and classical averages	For position and momentum, this gives $m,d^2\langle \hat{x}\rangle/dt^2 = -\langle \nabla V(\hat{x})\rangle$. When $V$ is sufficiently smooth over the wavepacket, the expectation values approximately obey Newton’s laws.
29	Dirac equation — αβ Hamiltonian (Dirac‑αβ) form	$i\hbar,\dfrac{\partial \psi}{\partial t} = \Big(c,\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} + \beta m c^2 + V\Big)\psi$	Relativistic Hamiltonian for spin‑½ particles	$\psi$ 4‑component spinor, $\hat{\mathbf{p}} = -i\hbar\nabla$, $m$ rest mass, $c$ speed of light. $\alpha_i$ and $\beta$ are $4\times 4$ Dirac matrices obeying ${\alpha_i,\alpha_j}=2\delta_{ij}$, ${\alpha_i,\beta}=0$, $\beta^2=1$. Reduces to free‑particle case when $V=0$.
30	Dirac equation — manifestly covariant form	$(i\hbar \gamma^\mu \partial_\mu - mc),\psi(x) = 0$	Lorentz‑covariant relativistic wave equation for spin‑½ fields	$\gamma^\mu$ Dirac gamma matrices satisfying ${\gamma^\mu,\gamma^\nu} = 2 g^{\mu\nu}$, $\partial_\mu = (\tfrac{1}{c}\partial_t,\nabla)$, $g^{\mu\nu}$ metric tensor. Relates to αβ form via $\gamma^0 = \beta$, $\gamma^i = \beta \alpha^i$. Often written as $(i\partial!!!/ - m)\psi=0$.
31	Quantum tunneling probability (rectangular barrier, WKB limit)	$T(E) \approx \exp(-2\kappa a),;\kappa = \sqrt{2m(V_0 - E)}/\hbar$	Approximate 1D barrier transmission; quantum devices, STM	Valid for $E<V_0$ and thick barrier $\kappa a \gg 1$. Prefactors from matching wavefunctions are omitted.
32	Quantum anharmonic oscillator	$\Big[-\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + \tfrac{1}{2}m\omega_0^2 x^2 + \lambda x^n\Big]\psi(x) = E\psi(x)$	Model for weakly nonlinear quantum systems	$\lambda$ coupling strength, $n\ge 3$ (often $n=3$ or $4$). Requires perturbation theory or numerics in general.
33	Landau–Lifshitz–Gilbert (LLG) equation	$\dfrac{\partial \mathbf{m}}{\partial t} = -\gamma,\mathbf{m}\times\mathbf{H}_{\mathrm{eff}} + \alpha,\mathbf{m}\times\dfrac{\partial \mathbf{m}}{\partial t}$	Magnetization dynamics in ferromagnets	$\mathbf{m} = \mathbf{M}/M_s$ unit magnetization, $\gamma$ gyromagnetic ratio (often negative for electrons), $\alpha$ Gilbert damping. $\mathbf{H}_{\mathrm{eff}}$ includes external, anisotropy, exchange, demagnetizing, and other effective fields.
34	ABCD/ transfer parameters	$\begin{bmatrix} V_1 \ I_1 \end{bmatrix} = \begin{bmatrix} A & B \ C & D \end{bmatrix} \begin{bmatrix} V_2 \ -I_2 \end{bmatrix}$	Two‑port networks; cascaded microwave/ optical components	Transfer (ABCD) matrix converts port‑2 voltage and current to port‑1. Cascaded networks multiply ABCD matrices.
35	Drift–diffusion current in semiconductors	$J_n = q n \mu_n E + q D_n,\dfrac{dn}{dx},\quad J_p = q p \mu_p E - q D_p,\dfrac{dp}{dx}$	Carrier transport in semiconductor devices	$J_n,J_p$ electron and hole current densities, $q$ elementary charge, $n,p$ carrier densities, $\mu_{n,p}$ mobilities, $D_{n,p}$ diffusion coefficients, $E$ electric field. Signs reflect negative electron and positive hole charge.
36	Effective field for LLG/ LLGS	$\mathbf{H}{\mathrm{eff}} = -\dfrac{1}{\mu_0 M_s},\dfrac{\delta F}{\delta \mathbf{m}} + \mathbf{H}{\mathrm{applied}} + \mathbf{H}_{\mathrm{demag}} + \cdots$	Micromagnetics; variational derivative of free energy	$F[\mathbf{m}]$ magnetic free‑energy functional (exchange, anisotropy, Zeeman, demagnetizing, etc.), $\mu_0$ vacuum permeability, $M_s$ saturation magnetization.
37	Scattering parameters (S‑parameters)	$\mathbf{b} = S,\mathbf{a}$, with $\mathbf{a} = (a_1,a_2)^{\mathsf{T}},;\mathbf{b} = (b_1,b_2)^{\mathsf{T}}$	Linear network characterization; RF, microwave, photonics	$S$ is $2\times 2$ matrix with elements $S_{ij}$ relating incident ($a_i$) and reflected/transmitted ($b_j$) wave amplitudes at ports.
38	Linear optical gain vs carrier density	$g(N) \approx \sigma_g (N - N_{\mathrm{tr}})$	Approximate constitutive relation; semiconductor lasers	$g$ modal gain, $N$ carrier density, $N_{\mathrm{tr}}$ transparency density, $\sigma_g$ differential gain or gain cross section. Valid near $N\approx N_{\mathrm{tr}}$.
39	Standard quantum limit of optical shot noise (amplitude and phase)	$S_{\mathrm{SQL}} = \dfrac{2 h \nu}{\bar{P}}$	Vacuum‑fluctuation shot‑noise limit of coherent light; amplitude and phase	$S_{\mathrm{SQL}}$ single‑sideband spectral density of relative noise (e.g., dBc/Hz), $h$ Planck constant, $\nu$ optical carrier frequency, $\bar{P}$ mean optical power. For coherent states, amplitude and phase quadratures have equal noise: $S_{\mathrm{SQL,amp}} = S_{\mathrm{SQL,phase}} = S_{\mathrm{SQL}}$.
40	Standard quantum limit of amplifier noise temperature (phase‑preserving)	$T_{\mathrm{N}} \ge \dfrac{h f}{2 k_{\mathrm{B}}}\quad\text{(equivalently }n_{\mathrm{add}} \ge \tfrac{1}{2}\text{)}$	Minimum noise temperature/added noise of linear, phase‑preserving quantum amplifier	$T_{\mathrm{N}}$ input‑referred noise temperature, $f$ signal frequency, $k_{\mathrm{B}}$ Boltzmann constant, $n_{\mathrm{add}}$ added quanta referred to the input. Equality is reached by a quantum‑limited amplifier; total input‑referred noise is then one quantum (signal vacuum + idler vacuum).
41	Free‑mass standard quantum limit (interferometric displacement)	$(\Delta z)_{\mathrm{SQL}} = \sqrt{\dfrac{2\hbar \tau}{m}}$	Continuous interferometric position measurement of a free mass; gravitational‑wave detectors	$m$ mass of each freely suspended test mass (e.g., interferometer end mirrors), $\tau$ measurement duration or integration time. Fundamental lower bound on root‑mean‑square position uncertainty from Heisenberg evolution of a free mass; often written for $z$ as arm‑length change.
42	Standard quantum limit of measurement noise (imprecision–back‑action)	$S_{xx}^{\mathrm{imp}}(\omega),S_{FF}^{\mathrm{ba}}(\omega) \ge \dfrac{\hbar^2}{4}$	General continuous linear quantum measurement; noise–back‑action tradeoff	$S_{xx}^{\mathrm{imp}}$ imprecision noise spectral density in measured observable (e.g., position), $S_{FF}^{\mathrm{ba}}$ back‑action force noise spectral density. Bound enforces that making one noise arbitrarily small drives the other large; minimizing total noise under this constraint yields specific SQLs.
43	Lattice Boltzmann equation (single‑relaxation BGK form)	$f_i(\mathbf{x}+\mathbf{c}_i \Delta t, t+\Delta t) - f_i(\mathbf{x}, t) = -\dfrac{\Delta t}{\tau},\big[f_i(\mathbf{x}, t) - f_i^{(eq)}(\mathbf{x}, t)\big]$	Mesoscopic fluid/ transport solver on discrete lattice	$f_i$ particle distribution along discrete velocity $\mathbf{c}_i$, $\tau$ relaxation time, $f_i^{(eq)}$ local equilibrium (typically low‑Mach expansion of Maxwell–Boltzmann). Macroscopic density and velocity follow from velocity moments and recover Navier–Stokes in the continuum limit.
44	Heisenberg limit for phase estimation	$\Delta\phi \gtrsim 1/N$	Ultimate quantum scaling for phase sensitivity	$N$ number of entangled particles, photons, or quanta in probe state. Beats shot‑noise limit $\Delta\phi \sim 1/\sqrt{N}$ using nonclassical states (e.g., N00N states).
45	LLGS equation with spin‑transfer torque	$\dfrac{\partial \mathbf{m}}{\partial t} = -\gamma,\mathbf{m}\times\mathbf{H}{\mathrm{eff}} + \alpha,\mathbf{m}\times\dfrac{\partial \mathbf{m}}{\partial t} + \boldsymbol{\tau}{\mathrm{STT}}$	Spintronics; current‑driven magnetization switching	Typical Slonczewski torque: $\boldsymbol{\tau}_{\mathrm{STT}} \propto \mathbf{m}\times(\mathbf{m}\times\mathbf{p})$, where $\mathbf{p}$ is spin‑polarization direction and the prefactor depends on current density, layer thickness, and material parameters.
46	Quantum lattice Boltzmann equation (QLB, schematic)	$\psi_i(\mathbf{x}+\mathbf{c}i \Delta t, t+\Delta t) = \sum_j U{ij}(\mathbf{x},t),\psi_j(\mathbf{x},t)$	Lattice‑based quantum evolution; discrete real‑space solver for Schrödinger/ Dirac	$\psi_i$ quantum amplitudes associated with discrete velocities or internal states, $U_{ij}$ unitary “collision” operator constructed so that the continuum limit reproduces Schrödinger or Dirac equations. Implementable on classical hardware and naturally suited to quantum computers as a quantum walk/ quantum circuit.
47	Fermi’s Golden Rule	$W_{i\to f} = \dfrac{2\pi}{\hbar},\big \lvert \langle f \lvert \hat{H}' \lvert i\rangle\big \lvert ^2,\rho(E_f)$	Transition rate in time‑dependent perturbation theory; quantum scattering, emission, absorption	$W_{i\to f}$ transition rate from initial state $\lvert i \rangle$ to final states near energy $E_f$, $\hat{H}'$ perturbing Hamiltonian, $\rho(E_f)$ density of final states at $E_f$. Assumes weak perturbation, long times, and quasi‑continuous spectrum.

Type of Gain	Definition (Linear Scale)	Definition (dB Scale)	Notes/ Context
Power Gain (RF/ Microwave)	$(G_p = \dfrac{P_\text{out}}{P_\text{in}})$	$(G_{p,\text{dB}} = 10\log_{10}(G_p) = 10\log_{10}!\left(\dfrac{P_\text{out}}{P_\text{in}}\right))$	Used when power flow is the key quantity (RF, microwave, link budgets). Factor 10 because dB is defined on power.
Voltage Gain (Low‑Frequency/ General Amplifier)	$(G_v = \dfrac{V_\text{out}}{V_\text{in}})$	$(G_{v,\text{dB}} = 20\log_{10}!\left \lvert G_v\right \lvert = 20\log_{10}!\left \lvert \dfrac{V_\text{out}}{V_\text{in}}\right \lvert )$	Common in audio and low‑frequency instrumentation. Factor 20 because $(P \propto \lvert V \lvert ^2)$ for fixed load impedance.
Current Gain	$(G_i = \dfrac{I_\text{out}}{I_\text{in}})$	$(G_{i,\text{dB}} = 20\log_{10}!\left \lvert G_i\right \lvert = 20\log_{10}!\left \lvert \dfrac{I_\text{out}}{I_\text{in}}\right \lvert )$	For true current amplifiers (current‑in, current‑out). Not used for transimpedance stages.
Transimpedance (Transresistance) Gain	$(Z_t = \dfrac{V_\text{out}}{I_\text{in}})$	$(Z_{t,\text{dB},\Omega} = 20\log_{10}!\left(\dfrac{ \lvert Z_t \lvert}{1~\Omega}\right) \approx 20\log_{10}!\left \lvert \dfrac{V_\text{out}}{I_\text{in}}\right \lvert)$	Key metric for photodiode and sensor front‑ends (current‑in, voltage‑out). Sometimes reported in dBΩ.

RF Power Gain
- Typically expressed in decibels (dB):
  
  $$G_{p,\mathrm{dB}} = 10 \log_{10}!\left(\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\right).$$
- Here $P_{\mathrm{out}}$ and $P_{\mathrm{in}}$ are average powers at the defined reference impedance (often $50~\Omega$ in RF systems).
- Widely used in radio-frequency and microwave engineering, antenna design, and link-budget calculations, where actual power transfer is the main concern.
Voltage Gain vs. Power Gain
- In low-frequency electronics (audio, instrumentation, op-amp circuits), voltage gain is often the primary specification, because signals are sensed as voltages.
- When the source and load impedances are equal and fixed,
  
  $$P \propto V^{2},$$
  
  so the decibel expression for voltage gain becomes
  
  $$G_{v,\mathrm{dB}} = 20 \log_{10}!\left(\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}\right),$$
  
  and this 20-log relationship is consistent with the 10-log power ratio under the equal-impedance assumption.
- If the impedances at input and output are different, $20 \log_{10}(V_{\mathrm{out}}/V_{\mathrm{in}})$ does not directly equal the power gain in dB; the impedance change must be accounted for if power gain is what you care about.
When to Use Which Formula
- Power-based gain (10 log): Use
  
  $$G_{\mathrm{dB}} = 10 \log_{10}!\left(\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\right)$$
  
  whenever you are comparing powers (e.g., RF power transfer, link budgets, amplifier power gain), regardless of the actual impedance values.
- Voltage-based gain (20 log): Use
  
  $$G_{\mathrm{dB}} = 20 \log_{10}!\left(\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}\right)$$
  
  when you are comparing voltage amplitudes (e.g., op-amp voltage gain, sensor front-ends), typically under the assumption that input and output are referred to the same characteristic impedance if you want the dB value to correspond to a power ratio.
- A similar 20-log rule applies for current gain:
  
  $$G_{i,\mathrm{dB}} = 20 \log_{10}!\left(\frac{I_{\mathrm{out}}}{I_{\mathrm{in}}}\right).$$
Alternate Notations
- In many textbooks and datasheets, you may see:
  - $G$ or $G_p$ for power gain,
  - $A_v$ for voltage gain,
  - $A_i$ for current gain,
  - $Z_t$ for transimpedance gain.
- The dB conversion always follows the same rule:
  - $10 \log_{10}$ for power ratios,
  - $20 \log_{10}$ for field or amplitude ratios (voltage, current), with attention to impedance if you want to interpret them as power ratios.
- Always check the context (what quantity is being compared, and at what impedance) to decide whether 10-log or 20-log is appropriate.

Wrap-up context

These gain definitions underpin electronic amplifier design, signal-chain analysis, and communications-system engineering. Correctly distinguishing between voltage, current, and power gain—and applying the proper dB formula with the right impedance assumptions—is essential for accurate specification and comparison of amplifiers.

Other key RF-amplifier parameters

Noise Figure (NF): Quantifies how much an amplifier degrades the signal-to-noise ratio; formally, NF is the ratio of input SNR to output SNR, often expressed in dB.
Bandwidth: The range of frequencies over which the amplifier meets its specified gain and performance (often defined between the $-3~\text{dB}$ gain points).
Linearity: Describes how well the amplifier preserves the proportionality between input and output; poor linearity leads to distortion and intermodulation products.

RF amplifiers are further categorized by operating class (Class A, B, AB, C, etc.) and by role (low-noise amplifier, driver amplifier, power amplifier, etc.), but all use these same gain and dB conventions.

Markdown script for the rendered tables above are available: click here

Field & potential equations
├─ Poisson (#5)
│  ├─ Laplace (#3)                [special case with ρ = 0]
│  ├─ Poisson–Boltzmann (#21)     [adds nonlinear screening by mobile ions]
│  └─ Green’s functions (#9)      [invert Poisson/Helmholtz with sources]
├─ Maxwell’s equations (#14)
│  ├─ Continuity equation (#15)           [charge/ probability conservation]
│  ├─ Wave equation (#2)                  [EM waves in vacuum/ media]
│  │  └─ Helmholtz equation (#18)         [time‑harmonic reduction of waves]
│  ├─ Poynting vector & theorem (#19)     [energy flow and conservation]
│  └─ Fourier transform (#6)              [k–ω domain solutions, dispersion]
└─ Geometric algebra (#17)
   └─ Compactly rewrites Maxwell (#14), Dirac (#29, #30), and wave (#2, #18) equations

Diffusion/ transport family
├─ Fick’s 1st law (#11) → Fick’s 2nd law (#12)
│  └─ Heat equation (#4)                  [same PDE with thermal parameters]
├─ Continuity equation (#15)
│  └─ Drift–diffusion currents (#35)
│     └─ Mean free path (#13)             [microscopic origin of transport coefficients]
├─ Navier–Stokes equation (#10)
│  └─ Lattice Boltzmann equation (#43)    [mesoscopic solver recovering Navier–Stokes]
└─ Poisson / Poisson–Boltzmann (#5, #21)
   └─ Coupled to drift–diffusion (#35) in semiconductor and electrolyte device models

Oscillators/ resonances
├─ Classical oscillators
│  ├─ Harmonic oscillator (#1)            [linear restoring force]
│  ├─ Anharmonic oscillator (#23)         [nonlinear corrections]
│  ├─ Resonant angular frequency (#16)    [ω₀ from parameters k, m or L, C]
│  ├─ Quality factor Q (#20)              [spectral sharpness/ damping]
│  └─ Scattering & transfer matrices (#37, #34)
│        [S‑parameters and ABCD matrices for resonant networks]
├─ Quantum oscillators & fields
│  ├─ Quantum harmonic oscillator (#26) and quantum anharmonic (#32)
│  ├─ Time‑dependent Schrödinger equation (#25)
│  │  └─ Time‑independent Schrödinger equation (#27) [eigenproblem for stationary states]
│  ├─ Dirac equations (#29, #30)          [relativistic spin‑½ dynamics]
│  ├─ Quantum tunneling probability (#31) [barrier penetration]
│  ├─ Quantum lattice Boltzmann (QLB) (#46)
│  │    [discrete streaming–collision realization of Schrödinger/ Dirac]
│  └─ Green’s functions (#9) in quantum propagation and scattering
│       └─ Fermi’s Golden Rule (#47)      [transition rates between eigenstates]
└─ Precision & noise (quantum limits)
   ├─ Shot‑noise current (#24)            [Poisson counting noise of charge/ photons]
   ├─ Optical shot‑noise SQL (#39)
   │    [vacuum‑fluctuation limit of coherent light, amplitude & phase quadratures]
   ├─ Amplifier SQL (#40)
   │    [phase‑preserving linear amplifier: n_add ≥ 1/2, T_n ≥ ħω/(2k_B)]
   ├─ Free‑mass SQL for displacement (#41)
   │    [Δx_SQL(τ) ≳ √(ħτ/ (2m)) for continuous tracking of a free mass]
   ├─ Measurement‑noise SQL product (#42)
   │    [S_xx^imp(ω) · S_FF^ba(ω) ≥ ħ²/4, imprecision–back‑action tradeoff]
   ├─ Heisenberg limit for phase estimation (#44)
   │    [Δφ ≳ 1/N, ultimate scaling with entangled probes]
   └─ Poynting/ Maxwell (#19, #14)
        [optical power flow and field dynamics in quantum‑limited interferometers]

Magnetization dynamics and spintronics
├─ Effective field H_eff (#36)
│  └─ LLG equation (#33)
│      └─ LLGS with spin‑transfer torque (STT) (#45)
└─ Coupled to:
   ├─ Poisson/ Maxwell (#5, #14)         [fields, currents, spin‑orbit torques]
   └─ Drift–diffusion (#35)               [spin‑polarized carrier transport]

Acronyms:

dB = decibel

NF = Noise Figure

PDE = Partial differential equation

EM = Electromagnetic / electromagnetism

LLG = Landau–Lifshitz–Gilbert equation

LLGS = Landau–Lifshitz–Gilbert–Slonczewski (LLG with spin‑transfer torque)

STT = Spin‑transfer torque

SQL = Standard quantum limit

QLB = Quantum lattice Boltzmann

QHO = Quantum harmonic oscillator

RF = Radio frequency

DOE = Design of Experiments.

GUM = Guide to the Expression of Uncertainty in Measurement.

LRT = Linear Response Theory.

CI = Confidence Interval (statistical coverage for an estimate).

PSD/SNR = Power Spectral Density/Signal-to-Noise Ratio (for noise-aware measurements).

BC = Boundary Conditions (geometry, constraints that shape curves).

PSD = Positive semidefinite

QFI = Quantum Fisher information

CPTP = Completely positive trace-preserving

RKHS = Reproducing kernel Hilbert space

QML = Quantum machine learning

QFT = Quantum field theory

GR = General relativity

SU(n) = Special unitary group of degree n

SO(n) = Special orthogonal group of degree n

U(1) = Unitary group of degree 1 (phase rotations)

References

0. Initial aspects

Hepburn, B. and Andersen, H., 2015. Scientific method. In: E.N. Zalta (ed.) The Stanford Encyclopedia of Philosophy (Fall 2015 ed.). Stanford, CA: Metaphysics Research Lab, Stanford University. Available at: https://plato.stanford.edu/entries/scientific-method/. ([Stanford Encyclopedia of Philosophy][1])
Jacob, S.L. and Goold, J., 2025. The response of a quantum system to a collision: an autonomous derivation of Kubo’s formula. Journal of Physics A: Mathematical and Theoretical (in press). Preprint available at: https://arxiv.org/abs/2505.03686. ([arXiv][2])
Watson, A.B., Margetis, D. and Luskin, M., 2023. Mathematical aspects of the Kubo formula for electrical conductivity with dissipation. Japan Journal of Industrial and Applied Mathematics, 40(3), pp.1765–1795. Preprint available at: https://arxiv.org/abs/2304.04303. ([SpringerLink][3])
NIST/SEMATECH, 2013. 4.3.1. What is design of experiments (DOE)? In: NIST/SEMATECH e‑Handbook of Statistical Methods. Gaithersburg, MD: National Institute of Standards and Technology. Available at: https://www.itl.nist.gov/div898/handbook/pmd/section3/pmd31.htm. ([NIST ITL][4])
Reiss, J. and Sprenger, J., 2020. Scientific objectivity. In: E.N. Zalta (ed.) The Stanford Encyclopedia of Philosophy (Winter 2020 ed.). Stanford, CA: Metaphysics Research Lab, Stanford University. Available at: https://plato.stanford.edu/entries/scientific-objectivity/. ([Stanford Encyclopedia of Philosophy][5])

1. Field & potential equations

Stanford University, Department of Mathematics, n.d. Green’s functions. Math 220B course notes, Stanford University. Available at: https://web.stanford.edu/class/math220b/handouts/greensfcns.pdf (Accessed 8 December 2025). ([Stanford University][1]) – Covers Green’s functions for Poisson, Laplace, Helmholtz and related PDEs → Poisson (#5), Laplace (#3), Green’s functions (#9), Helmholtz (#18).
Gray, F., 2018. ‘Nonlinear electrostatics: the Poisson–Boltzmann equation.’ European Journal of Physics, 39(5), 053001. Preprint available at: https://arxiv.org/abs/1808.08338 (Accessed 8 December 2025). – Focused on Poisson–Boltzmann theory and screening, with clear derivation and physical interpretation → Poisson–Boltzmann (#21), coupling to Poisson (#5).
Wikipedia, 2024. ‘Poisson’s equation.’ Wikipedia, the free encyclopedia. Available at: https://en.wikipedia.org/wiki/Poisson%27s_equation (Accessed 8 December 2025). ([DAMP][2]) – Concise summary of Poisson’s equation, boundary conditions, and applications → Poisson (#5) and links to Laplace and Green’s‑function solutions.
Wikipedia, 2024. ‘Maxwell’s equations.’ Wikipedia, the free encyclopedia. Available at: https://en.wikipedia.org/wiki/Maxwell%27s_equations (Accessed 8 December 2025). – Good overview of the differential and integral forms, wave equation, and Poynting vector → Maxwell (#14), wave equation (#2), Poynting vector & theorem (#19).
Hestenes, D., 2003. ‘Oersted Medal Lecture 2002: Reforming the mathematical language of physics.’ American Journal of Physics, 71(2), pp.104–121. Author’s OA version available at: http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf (Accessed 8 December 2025). – Introduces geometric (Clifford) algebra as a unifying language for Maxwell, Dirac, and wave equations → Geometric algebra (#17), Maxwell (#14), Dirac (#29, #30).

2. Diffusion/ transport family

Wikipedia, 2024. ‘Fick’s laws of diffusion.’ Wikipedia, the free encyclopedia. Available at: https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion (Accessed 8 December 2025). ([Wikipedia][3]) – Standard reference for Fick’s first and second laws, diffusion equation, and relation to random walks → Fick’s laws (#11, #12), heat/diffusion equation (#4).
Gentle introduction via PDEs: Stanford University, Department of Mathematics, n.d. The heat equation. Course notes (Math PDE sequence). Available at: https://web.stanford.edu/class/math220b/handouts/HeatEqn.pdf (Accessed 8 December 2025). ([arXiv][4]) – Derives the heat equation as a prototype parabolic PDE and links it to diffusion → Heat equation (#4) and general diffusion/transport perspective.
Jung, I. & Selberherr, S., 2013. ‘Determination of drift–diffusion parameters for semiconductor device simulation.’ Journal of Computational Electronics, 12(4), pp.701–708. Author OA version: https://arxiv.org/abs/1305.4861 (Accessed 8 December 2025). – Gives a device‑physics take on drift–diffusion transport and parameter extraction → Drift–diffusion currents (#35), mean free path and transport coefficients (#13).
Wikipedia, 2024. ‘Navier–Stokes equations.’ Wikipedia, the free encyclopedia. Available at: https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations (Accessed 8 December 2025). – High‑level description of Navier–Stokes, derivation from conservation laws, and various limits → Navier–Stokes (#10), connection to diffusion and continuity (#15).
Hosseini, S.A., Boivin, P., Thévenin, D. & Karlin, I.V., 2024. ‘Lattice Boltzmann methods for combustion applications.’ Progress in Energy and Combustion Science, 102, 101140. Open‑access article at: https://doi.org/10.1016/j.pecs.2023.101140 (Accessed 8 December 2025). – Broad review including fundamentals of the lattice Boltzmann method and its continuum (Navier–Stokes) limit → Lattice Boltzmann (#43) and its relation to Navier–Stokes (#10).

3. Oscillators, quantum fields & tunneling

Essler, F.H.L., 2023. Lecture Notes for Quantum Mechanics. University of Oxford. Available at: https://www-thphys.physics.ox.ac.uk/people/FabianEssler/QM/QM2022.pdf (Accessed 8 December 2025). – Systematic derivation of time‑dependent and time‑independent Schrödinger equations, harmonic oscillator, and perturbation theory → TDSE (#25), TISE (#27), QHO (#26), anharmonic oscillator (#32).
Chew, W.C., 2013. Quantum Mechanics Made Simple: Lecture Notes. Purdue University. Available at: https://engineering.purdue.edu/wcchew/course/QMAll20130917.pdf (Accessed 8 December 2025). – Gentle yet thorough notes emphasizing physical intuition and simple models such as harmonic and anharmonic oscillators → QHO (#26), anharmonic potentials (#32), Ehrenfest connections (#28).
Williams, B., 2017. ‘Quantum harmonic oscillator and generalized Hermite polynomials.’ arXiv preprint arXiv:1701.02767. Available at: https://arxiv.org/abs/1701.02767 (Accessed 8 December 2025). ([arXiv][5]) – Focuses mathematically on QHO eigenfunctions and generalizations → Deepens QHO (#26) and connects to more exotic oscillator models (#32).
Haouam, I., 2023. ‘Ehrenfest’s theorem for the Dirac equation in curved spacetime.’ arXiv preprint arXiv:2309.16360. Available at: https://arxiv.org/abs/2309.16360 (Accessed 8 December 2025). – Explicitly treats Ehrenfest’s theorem in relativistic quantum mechanics → Ehrenfest’s theorem (#28), Dirac equations (#29, #30).
Makris, N., 2024. ‘A real‑valued description of quantum mechanics with classical channels.’ arXiv preprint arXiv:2406.05484. Available at: https://arxiv.org/abs/2406.05484 (Accessed 8 December 2025). – Re‑derives Schrödinger‑type dynamics in a real‑valued framework and discusses classical/quantum correspondence → Conceptual support around Schrödinger equations (#25, #27) and quantum–classical links near Ehrenfest’s theorem (#28).
Dereziński, J. & Gérard, C., 2017. Mathematics of Quantization and Quantum Fields. Lecture Notes in Physics 2157 (open version). Chapter on perturbation theory and Fermi’s Golden Rule. OA PDF: https://ndl.ethernet.edu.et/bitstream/123456789/63327/1/157.pdf (Accessed 8 December 2025). – Rigorous treatment of quantum fields and Fermi’s Golden Rule → Fermi’s Golden Rule (#47), Green’s functions in quantum propagation (#9).
Schmeissner, R., Thiel, V., Jacquard, C., Fabre, C. & Treps, N., 2014. ‘Analysis and filtering of phase noise in an optical frequency comb at the quantum limit to improve timing measurements.’ Optics Letters, 39(12), pp.3603–3606. OA arXiv version: https://arxiv.org/abs/1401.3528 (Accessed 8 December 2025). – Real experimental context where quantum‑limited noise, oscillators, and comb modes meet → Oscillators/resonances, precision & noise branch, especially optical shot‑noise SQL (#39).
Wang, K. et al., 2019. Multidimensional photonics in synthetic lattices. PhD thesis, Australian National University. OA thesis: https://openresearch-repository.anu.edu.au/bitstreams/29581b51-042a-4d07-ab86-132b92be082a/download (Accessed 8 December 2025). – Explores photonic lattices, band structures and synthetic dimensions, with tight links to wave equations, resonances and quantum walks → Bridges oscillator / resonator networks to lattice models and QLB (#46).
Succi, S., 2015. ‘Lattice Boltzmann 2038.’ EPL (Europhysics Letters), 109, 50001. OA version: https://doi.org/10.1209/0295-5075/109/50001 (Accessed 8 December 2025). – Visionary overview of lattice Boltzmann and its quantum variants → Connects classical LBM (#43) and quantum lattice Boltzmann (#46) to future quantum simulators.

4. Precision, quantum noise & measurement limits

Caves, C.M., 1981. ‘Quantum‑mechanical noise in an interferometer.’ Physical Review D, 23(8), pp.1693–1708. OA copy: https://www.rle.mit.edu/cua_pub/8.422/Reading%20Material/caves_PRD.pdf (Accessed 8 December 2025). – Classic derivation of the interferometric standard quantum limit and introduction of squeezed states → Free‑mass SQL (#41), optical shot‑noise SQL (#39), Heisenberg vs. SQL in interferometers (#44).
Clerk, A.A., Devoret, M.H., Girvin, S.M., Marquardt, F. & Schoelkopf, R.J., 2010. ‘Introduction to quantum noise, measurement, and amplification.’ Reviews of Modern Physics, 82(2), pp.1155–1208. OA PDF: https://clerkgroup.uchicago.edu/PDFfiles/RMP2010.pdf (Accessed 8 December 2025). – Canonical modern review on quantum noise spectra, linear response, and amplifier limits → Amplifier SQL (#40), measurement‑noise product SQL (#42), Heisenberg limit (#44).
King, S.K., 1996. ‘Quantum mechanical noise in a Michelson interferometer.’ NASA Technical Memorandum 107585. OA report: https://ntrs.nasa.gov/api/citations/19960025005/downloads/19960025005.pdf (Accessed 8 December 2025). – Applies quantum‑noise ideas to real interferometer design → Practical elaboration of free‑mass displacement SQL (#41) and optical shot noise (#39).
Braginsky, V.B. & Khalili, F.Y., 1992. Quantum Measurement. Cambridge University Press. OA individual chapters collected in: Measurements in Quantum Mechanics (InTechOpen). See, e.g.,: Summarized OA volume Measurements in Quantum Mechanics, 2012. Available at: https://www.issp.ac.ru/ebooks/books/open/Measurements_in_Quantum_Mechanics.pdf (Accessed 8 December 2025). – Foundational monograph on quantum measurements, back‑action, and SQL in oscillators and interferometers → Deep background for all SQL entries (#39–#42) and Heisenberg limit (#44).
Balmaseda, A., 2022. Quantum Control at the Boundary. PhD dissertation (preprint). arXiv preprint arXiv:2201.05480. Available at: https://arxiv.org/abs/2201.05480 (Accessed 8 December 2025). – Dissertational perspective on controlling quantum systems via boundary conditions; discusses limits to control and measurement → Conceptual complement to your “precision & noise” subtree and quantum‑limited control schemes.

5. Magnetization dynamics and spintronics

Lakshmanan, M., 2011. ‘The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview.’ In: Nonlinear Dynamics: Integrability, Chaos and Patterns. OA version: https://arxiv.org/pdf/1101.1005.pdf (Accessed 8 December 2025). – Pedagogical overview of LLG, solitons, and nonlinear dynamics → Effective field (H_\mathrm{eff}) (#36), LLG equation (#33), links to nonlinear oscillators (#23).
Xu, F. et al., 2023. ‘Unified framework of the microscopic Landau–Lifshitz–Gilbert equation and its application to skyrmion dynamics.’ arXiv preprint arXiv:2310.08807. Available at: https://arxiv.org/abs/2310.08807 (Accessed 8 December 2025). – Derives LLG from nonequilibrium Green’s functions, giving microscopic expressions for damping and torques → Refines effective‑field and damping structure for LLG/LLGS (#33, #36, #45).
Ralph, D.C. & Stiles, M.D., 2008. ‘Spin transfer torques.’ Journal of Magnetism and Magnetic Materials, 320(7), pp.1190–1216. OA arXiv version: https://arxiv.org/abs/0711.4608 (Accessed 8 December 2025). – Tutorial review of spin‑transfer torque physics in multilayers → LLGS with STT (#45), coupling of drift–diffusion (#35) to magnetization dynamics (#33).
Saradzhev, F.M., Khanna, F.C., Kim, S.P. & de Montigny, M., 2006. ‘General form of magnetization damping: magnetization dynamics of a spin system evolving nonadiabatically and out of equilibrium.’ arXiv preprint arXiv:cond-mat/0609431. Available at: https://arxiv.org/abs/cond-mat/0609431 (Accessed 8 December 2025). – Extends LLG to more general damping forms using quantum density‑matrix methods → Links your effective‑field and damping functionals (#36) to quantum open‑system models.
Wikipedia, 2024. ‘Landau–Lifshitz–Gilbert equation.’ Wikipedia, the free encyclopedia. Available at: https://en.wikipedia.org/wiki/Landau%E2%80%93Lifshitz%E2%80%93Gilbert_equation (Accessed 8 December 2025). – Compact overview of LLG, LLGS and spin‑torque extensions, with references to original literature → LLG (#33), LLGS with STT (#45), effective field construction (#36).

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