quantum-physics-ai999x — offline, verified quantum physics solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Physical Sciences · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

quantum-physics-ai999x is an offline, deterministic Python solver that answers quantum physics questions with verified results from 75 certified capabilities, gives 376 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

quantum-physics-ai999x is a compact, deterministic quantum physics engine for Python. It reads a question in plain English (or a typed call), picks a certified quantum physics capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified quantum physics capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 376 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: special relativity, photons and matter waves, the hydrogen atom and angular momentum, quantum wells and tunnelling, spin, qubits and quantum gates, nuclear and particle physics, statistical physics.

Model overview

Property	Value
Package	quantum-physics-ai999x 1.0.0 (import quantum_physics_ai999x)
Type	Deterministic neuro-symbolic quantum physics engine (rule-based reading + certified symbolic capabilities)
Capabilities	75 certified procedures
Knowledge pack	version 1.0.1
Golden answers	376 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install quantum-physics-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/quantum-physics-ai999x/resolve/main/dist/quantum_physics_ai999x-1.0.0-py3-none-any.whl

import quantum_physics_ai999x as quantum_physics

quantum_physics.ask("Lorentz factor: v = 0.8 c").show()
# -> SOLVED — gamma = 1.66667, beta = 0.8, gamma_minus_1 = 0.666667  (lorentz_factor)
quantum_physics.ask("Hydrogen transition: Z = 1, n_upper = 3, n_lower = 2").show()
# -> SOLVED — E_photon_eV = 1.88968, lam = 6.56112e-07, lambda_nm = 656.112, f = 4.56922e+14  (hydrogen_transition)
quantum_physics.ask("Photoelectric effect: lambda = 250 nm, W = 4.3 eV").show()
# -> SOLVED — E_photon_eV = 4.95937, K_max_eV = 0.659368, V_stop = 0.659368, lambda_threshold = 2.88335e-07  (photoelectric_effect)
quantum_physics.solve("two_qubit_circuit", start="00", gates="H0 CNOT01").show()
# -> SOLVED — p00 = 0.5, p01 = 0, p10 = 0, p11 = 0.5  (two_qubit_circuit)
quantum_physics.ask("Barrier transmission: E = 1 eV, V0 = 2 eV, a = 0.5 nm, m = 9.109e-31 kg").show()
# -> SOLVED — T = 0.0235496, R = 0.97645, kappa_a = 2.56153, regime = tunnelling (E < V0)  (barrier_transmission)
quantum_physics.ask("Who created you?").message
# -> quantum-physics-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: quantum-physics-ai999x "<question>", quantum-physics-ai999x --capabilities. The outputs above were produced by running this exact code on quantum-physics-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

75 certified procedures. A few of them:

• Procedures: Age from the remaining fraction (radioactive dating); Bloch-sphere rotation of a single-qubit gate sequence; Bose-Einstein condensation temperature; Bose-Einstein occupation of a mode; Bound states of a finite square well; CHSH Bell-test value for the singlet state; Centre-of-mass energy of a fixed-target collision; Chemical potential of an ideal Fermi gas; Clebsch-Gordan coefficient; Compton scattering; Coulomb barrier between two nuclei; Debye heat capacity of a solid; Decay of a named radionuclide; Energy and photon density of blackbody radiation; Energy level of a hydrogen-like atom (Bohr model); Energy levels of a particle in a box; Fermi energy of a free-electron gas; Fermi energy of a metal from its density; Fermi-Dirac occupation of a state; Fidelity and trace distance of two qubit states; Fraction of blackbody radiation in a wavelength band; Hydrogen-like transition with the reduced-mass correction; Invariant mass of two particles; K-alpha X-ray line (Moseley's law) …

quantum_physics_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 75 formulas

Show every formula, by topic

Angular momentum / addition of angular momenta

Clebsch-Gordan coefficient

$$⟨j_1{m}_1;j_2{m}_2\mathrm{\mid }JM⟩\backslash langle\; j\_1\; m\_1;\; j\_2\; m\_2\; \backslash vert\; J\; M\backslash rangle$$

Wigner 3j symbol

Hydrogen atom / X-ray spectra

K-alpha X-ray line (Moseley's law)

$$E_{K\alpha }=\frac{3}{4}{R}_{\mathrm{\infty }}hc(Z-1{)}^{2}E\_\{K\backslash alpha\}\; =\; \backslash tfrac\{3\}\{4\}\; R\_\backslash infty\; h\; c\backslash ,(Z\; -\; 1)^2$$

Hydrogen atom / Zeeman effect

Landé g factor and anomalous Zeeman shift

$$g_J=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)},\mathrm{\Delta }E=g_J{\mu }_{B}B{m}_{J}g\_J\; =\; 1\; +\; \backslash frac\{J(J+1)\; +\; S(S+1)\; -\; L(L+1)\}\{2J(J+1)\},\backslash quad\; \backslash Delta\; E\; =\; g\_J\; \backslash mu\_B\; B\; m\_J$$

Hydrogen atom / energy levels

Energy level of a hydrogen-like atom (Bohr model)

$$E_n=-\frac{Z^2{R}_{\mathrm{\infty }}hc}{n^2}\approx -\frac{13.6\mathrm{e}\mathrm{V}{Z}^2}{n^2}E\_n\; =\; -\backslash frac\{Z^2\; R\_\backslash infty\; h\; c\}\{n^2\}\; \backslash approx\; -\backslash frac\{13.6\backslash \; \backslash mathrm\{eV\}\backslash ,Z^2\}\{n^2\}$$

Hydrogen atom / spectral lines

Photon from a hydrogen-like transition (Rydberg formula)

$$\frac{1}{\lambda }=R_{\mathrm{\infty }}{Z}^2(\frac{1}{n_{\mathrm{\ell }}^2}-\frac{1}{n_u^2})\backslash frac\{1\}\{\backslash lambda\}\; =\; R\_\backslash infty\; Z^2\backslash left(\backslash frac\{1\}\{n\_\backslash ell^2\}\; -\; \backslash frac\{1\}\{n\_u^2\}\backslash right)$$

Hydrogen atom / spectral lines and isotope shift

Hydrogen-like transition with the reduced-mass correction

$$\frac{1}{\lambda }=\frac{\mu }{m_e}{R}_{\mathrm{\infty }}{Z}^2(\frac{1}{n_{\mathrm{\ell }}^2}-\frac{1}{n_u^2})\backslash frac\{1\}\{\backslash lambda\}\; =\; \backslash frac\{\backslash mu\}\{m\_e\}\; R\_\backslash infty\; Z^2\backslash left(\backslash frac\{1\}\{n\_\backslash ell^2\}\; -\; \backslash frac\{1\}\{n\_u^2\}\backslash right)$$

Hydrogen atom / wavefunctions

Probability of finding the electron within a radius

Radial expectation values of a hydrogen-like state

$$⟨r⟩=\frac{a_0}{2Z}[3n^2-l(l+1)]\backslash langle\; r\backslash rangle\; =\; \backslash frac\{a\_0\}\{2Z\}\backslash left[3n^2\; -\; l(l+1)\backslash right]$$

Nuclear and particle physics / binding energy

Semi-empirical mass formula (liquid-drop binding energy)

$$B=a_{V}A-a_S{A}^{2\mathrm{/}3}-a_C\frac{Z(Z-1)}{A^{1\mathrm{/}3}}-a_A\frac{(A-2Z{)}^2}{A}+\delta (A,Z)B\; =\; a\_V\; A\; -\; a\_S\; A^\{2/3\}\; -\; a\_C\backslash frac\{Z(Z-1)\}\{A^\{1/3\}\}\; -\; a\_A\backslash frac\{(A-2Z)^2\}\{A\}\; +\; \backslash delta(A,Z)$$

Nuclear and particle physics / decay kinematics and Q value

Kinematics of a two-body decay

$$p=\frac{\sqrt{[M^2-(m_1+m_2{)}^2][M^2-(m_1-m_2{)}^2]}}{2M}p\; =\; \backslash frac\{\backslash sqrt\{\backslash left[M^2\; -\; (m\_1\; +\; m\_2)^2\backslash right]\backslash left[M^2\; -\; (m\_1\; -\; m\_2)^2\backslash right]\}\}\{2M\}$$

Nuclear physics / decay chains and equilibrium

Parent-daughter decay chain (Bateman)

$$N_D(t)=N_0\frac{{\lambda }_P}{{\lambda }_D-{\lambda }_P}(e^{-{\lambda }_{P}t}-e^{-{\lambda }_{D}t})N\_D(t)\; =\; N\_0\backslash frac\{\backslash lambda\_P\}\{\backslash lambda\_D\; -\; \backslash lambda\_P\}\backslash left(e^\{-\backslash lambda\_P\; t\}\; -\; e^\{-\backslash lambda\_D\; t\}\backslash right)$$

Nuclear physics / nuclear size and the Coulomb barrier

Coulomb barrier between two nuclei

$$V_C=\frac{Z_1{Z}_2{e}^2}{4\pi {\epsilon }_0{r}_0(A_1^{1\mathrm{/}3}+A_2^{1\mathrm{/}3})}V\_C\; =\; \backslash frac\{Z\_1\; Z\_2\; e^2\}\{4\backslash pi\backslash varepsilon\_0\backslash ,r\_0\backslash left(A\_1^\{1/3\}\; +\; A\_2^\{1/3\}\backslash right)\}$$

Nuclear physics / radioactive dating

Age from the remaining fraction (radioactive dating)

$$t=\frac{T_{1\mathrm{/}2}}{\ln 2}\ln \frac{N_0}{N}t\; =\; \backslash frac\{T\_\{1/2\}\}\{\backslash ln\; 2\}\backslash ln\backslash frac\{N\_0\}\{N\}$$

Nuclear physics / radioactive decay

Radioactive decay law

$$N(t)=N_0{2}^{-t\mathrm{/}{T}_{1\mathrm{/}2}}=N_0{e}^{-\lambda t},A=\lambda NN(t)\; =\; N\_0\backslash ,2^\{-t/T\_\{1/2\}\}\; =\; N\_0\; e^\{-\backslash lambda\; t\},\backslash quad\; A\; =\; \backslash lambda\; N$$

Nuclear physics / radioactive decay of real nuclides

Decay of a named radionuclide

$$\frac{N}{N_0}=2^{-t\mathrm{/}{T}_{1\mathrm{/}2}}\backslash frac\{N\}\{N\_0\}\; =\; 2^\{-t/T\_\{1/2\}\}$$

Nuclear physics / reactions, Q value and threshold

Q value and threshold energy of a nuclear reaction

$$Q=(m_a+m_X-m_Y-m_b)c^2,K_{th}=-Q\frac{m_a+m_X+m_Y+m_b}{2m_X}Q\; =\; (m\_a\; +\; m\_X\; -\; m\_Y\; -\; m\_b)c^2,\backslash quad\; K\_\{th\}\; =\; -Q\backslash ,\backslash frac\{m\_a\; +\; m\_X\; +\; m\_Y\; +\; m\_b\}\{2\; m\_X\}$$

Particle physics / lifetimes and time dilation

Mean decay length of a relativistic unstable particle

$$L=\beta \gamma c\tau =\frac{p}{mc}c\tau L\; =\; \backslash beta\backslash gamma\; c\backslash tau\; =\; \backslash frac\{p\}\{mc\}\backslash ,c\backslash tau$$

Photons and matter waves / Compton effect

Compton scattering

$${\lambda }^{\mathrm{\prime }}-\lambda =\frac{h}{m_{e}c}(1-\cos \theta )\backslash lambda\text{'}\; -\; \backslash lambda\; =\; \backslash frac\{h\}\{m\_e\; c\}\backslash left(1\; -\; \backslash cos\backslash theta\backslash right)$$

Photons and matter waves / coherent states of light

Photon-number distribution of a coherent state

$$P(n)=e^{-\mathrm{\mid }\alpha {\mathrm{\mid }}^2}\frac{\mathrm{\mid }\alpha {\mathrm{\mid }}^{2n}}{n!}P(n)\; =\; e^\{-\backslash vert\; \backslash alpha\backslash vert\; ^2\}\backslash frac\{\backslash vert\; \backslash alpha\backslash vert\; ^\{2n\}\}\{n!\}$$

Photons and matter waves / de Broglie wavelength

de Broglie wavelength of a moving particle

$$\lambda =\frac{h}{p}=\frac{h}{mv}\backslash lambda\; =\; \backslash frac\{h\}\{p\}\; =\; \backslash frac\{h\}\{m\; v\}$$

de Broglie wavelength of an accelerated particle

$$\lambda =\frac{h}{p},p=\frac{1}{c}\sqrt{K(K+2m{c}^2)}\backslash lambda\; =\; \backslash frac\{h\}\{p\},\backslash quad\; p\; =\; \backslash frac\{1\}\{c\}\backslash sqrt\{K(K\; +\; 2mc^2)\}$$

Photons and matter waves / photoelectric effect

Photoelectric effect

$$K_{max}=\frac{hc}{\lambda }-W=e{V}_{stop}K\_\{max\}\; =\; \backslash frac\{hc\}\{\backslash lambda\}\; -\; W\; =\; e\; V\_\{stop\}$$

Photons and matter waves / photon energy

Photon energy from wavelength

$$E=hf=\frac{hc}{\lambda },p=\frac{h}{\lambda }E\; =\; h\; f\; =\; \backslash frac\{hc\}\{\backslash lambda\},\backslash quad\; p\; =\; \backslash frac\{h\}\{\backslash lambda\}$$

Photons and matter waves / uncertainty principle

Minimum momentum uncertainty (Heisenberg)

$$\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\mathrm{\hslash }}{2}\backslash Delta\; x\backslash ,\backslash Delta\; p\; \backslash ge\; \backslash frac\{\backslash hbar\}\{2\}$$

Quantum gates / gates as rotations

Bloch-sphere rotation of a single-qubit gate sequence

$$U\simeq e^{-i\theta \widehat{n}\cdot \vec{\sigma }\mathrm{/}2}=\cos \frac{\theta }{2}I-i\sin \frac{\theta }{2}\widehat{n}\cdot \vec{\sigma }U\; \backslash simeq\; e^\{-i\backslash theta\backslash ,\backslash hat\; n\backslash cdot\backslash vec\backslash sigma/2\}\; =\; \backslash cos\backslash tfrac\{\backslash theta\}\{2\}\; I\; -\; i\backslash sin\backslash tfrac\{\backslash theta\}\{2\}\backslash ,\backslash hat\; n\backslash cdot\backslash vec\backslash sigma$$

Quantum gates / quantum algorithms

Success probability of Grover's search

$$P_k={\sin }^2\text{\hspace{0.17em}⁣}((2k+1)\theta ),\sin \theta =\sqrt{M\mathrm{/}N}P\_k\; =\; \backslash sin^2\backslash !\backslash big((2k+1)\backslash theta\backslash big),\backslash quad\; \backslash sin\backslash theta\; =\; \backslash sqrt\{M/N\}$$

Quantum gates / single-qubit gates

State of a qubit after a sequence of gates

$$\mathrm{\mid }\psi ⟩=U_k\cdots U_2{U}_1\mathrm{\mid }{\psi }_0⟩\backslash vert\; \backslash psi\backslash rangle\; =\; U\_k\; \backslash cdots\; U\_2\; U\_1\; \backslash vert\; \backslash psi\_0\backslash rangle$$

Quantum gates / two-qubit gates and Bell states

Two-qubit circuit: probabilities and entanglement

$$\mathrm{\mid }{\mathrm{\Phi }}^{+}⟩={\mathrm{C}\mathrm{N}\mathrm{O}\mathrm{T}}_{01}(H\otimes I)\mathrm{\mid }00⟩=\frac{1}{\sqrt{2}}(\mathrm{\mid }00⟩+\mathrm{\mid }11⟩)\backslash vert\; \backslash Phi^+\backslash rangle\; =\; \backslash mathrm\{CNOT\}\_\{01\}(H\backslash otimes\; I)\backslash vert\; 00\backslash rangle\; =\; \backslash tfrac\{1\}\{\backslash sqrt2\}(\backslash vert\; 00\backslash rangle\; +\; \backslash vert\; 11\backslash rangle)$$

Quantum wells / expectation values and uncertainty

Position and momentum spread in a box

$$\mathrm{\Delta }x=L\sqrt{\frac{1}{12}-\frac{1}{2n^2{\pi }^2}},\mathrm{\Delta }p=\frac{n\pi \mathrm{\hslash }}{L}\backslash Delta\; x\; =\; L\backslash sqrt\{\backslash frac\{1\}\{12\}\; -\; \backslash frac\{1\}\{2n^2\backslash pi^2\}\},\backslash quad\; \backslash Delta\; p\; =\; \backslash frac\{n\backslash pi\backslash hbar\}\{L\}$$

Quantum wells / finite square well

Bound states of a finite square well

$$z\tan z=\sqrt{z_0^2-z^2},-z\cot z=\sqrt{z_0^2-z^2},z_0=\frac{a}{2\mathrm{\hslash }}\sqrt{2m{V}_0}z\backslash tan\; z\; =\; \backslash sqrt\{z\_0^2\; -\; z^2\},\backslash quad\; -z\backslash cot\; z\; =\; \backslash sqrt\{z\_0^2\; -\; z^2\},\backslash quad\; z\_0\; =\; \backslash frac\{a\}\{2\backslash hbar\}\backslash sqrt\{2mV\_0\}$$

Quantum wells / harmonic oscillator

Quantum harmonic oscillator levels

$$E_n=(n+\frac{1}{2})\mathrm{\hslash }\omega E\_n\; =\; \backslash left(n\; +\; \backslash tfrac12\backslash right)\backslash hbar\backslash omega$$

Quantum wells / harmonic oscillator and tunnelling

Probability beyond the classical turning points (harmonic oscillator)

$$P=1-{\int }_{-\sqrt{2n+1}}^{\sqrt{2n+1}}\mathrm{\mid }{\psi }_n(\xi ){\mathrm{\mid }}^{2}d\xi P\; =\; 1\; -\; \backslash int\_\{-\backslash sqrt\{2n+1\}\}^\{\backslash sqrt\{2n+1\}\}\; \backslash vert\; \backslash psi\_n(\backslash xi)\backslash vert\; ^2\backslash ,d\backslash xi$$

Quantum wells / infinite square well

Energy levels of a particle in a box

$$E_n=\frac{n^2{h}^2}{8m{L}^2}E\_n\; =\; \backslash frac\{n^2\; h^2\}\{8\; m\; L^2\}$$

Photon from a particle-in-a-box transition

$$\mathrm{\Delta }E=\frac{(n_u^2-n_{\mathrm{\ell }}^2)h^2}{8m{L}^2}=\frac{hc}{\lambda }\backslash Delta\; E\; =\; \backslash frac\{(n\_u^2\; -\; n\_\backslash ell^2)\backslash ,h^2\}\{8\; m\; L^2\}\; =\; \backslash frac\{hc\}\{\backslash lambda\}$$

Probability of finding a particle in part of a box

$$P={\int }_{x_1}^{x_2}\frac{2}{L}{\sin }^2\frac{n\pi x}{L}dxP\; =\; \backslash int\_\{x\_1\}^\{x\_2\}\; \backslash frac\{2\}\{L\}\backslash sin^2\backslash frac\{n\backslash pi\; x\}\{L\}\backslash ,dx$$

Quantum wells / potential step

Reflection and transmission at a potential step

$$R={\left(\frac{k_1-k_2}{k_1+k_2}\right)}^2,T=\frac{4k_1{k}_2}{(k_1+k_2{)}^2},\frac{k_2}{k_1}=\sqrt{\frac{E-V_0}{E}}R\; =\; \backslash left(\backslash frac\{k\_1\; -\; k\_2\}\{k\_1\; +\; k\_2\}\backslash right)^2,\backslash quad\; T\; =\; \backslash frac\{4k\_1k\_2\}\{(k\_1\; +\; k\_2)^2\},\backslash quad\; \backslash frac\{k\_2\}\{k\_1\}\; =\; \backslash sqrt\{\backslash frac\{E\; -\; V\_0\}\{E\}\}$$

Quantum wells / tunnelling

Transmission through a rectangular barrier

$$T={[1+\frac{V_0^2{\sinh }^2(\kappa a)}{4E(V_0-E)}]}^{-1},\kappa =\frac{\sqrt{2m(V_0-E)}}{\mathrm{\hslash }}T\; =\; \backslash left[1\; +\; \backslash frac\{V\_0^2\backslash sinh^2(\backslash kappa\; a)\}\{4E(V\_0\; -\; E)\}\backslash right]^\{-1\},\backslash quad\; \backslash kappa\; =\; \backslash frac\{\backslash sqrt\{2m(V\_0\; -\; E)\}\}\{\backslash hbar\}$$

Quantum wells / tunnelling (WKB)

WKB tunnelling through a triangular barrier (field emission)

$$T\approx \exp \text{\hspace{0.17em}⁣}(-\frac{4\sqrt{2m_e}{W}^{3\mathrm{/}2}}{3e\mathrm{\hslash }F})T\; \backslash approx\; \backslash exp\backslash !\backslash left(-\backslash frac\{4\backslash sqrt\{2m\_e\}\backslash ,W^\{3/2\}\}\{3\; e\backslash hbar\; F\}\backslash right)$$

Special relativity / Lorentz factor

Lorentz factor

$$\gamma =\frac{1}{\sqrt{1-v^2\mathrm{/}{c}^2}}\backslash gamma\; =\; \backslash frac\{1\}\{\backslash sqrt\{1\; -\; v^2/c^2\}\}$$

Special relativity / aberration

Relativistic aberration of light

$$\cos {\theta }^{\mathrm{\prime }}=\frac{\cos \theta -\beta }{1-\beta \cos \theta }\backslash cos\backslash theta\text{'}\; =\; \backslash frac\{\backslash cos\backslash theta\; -\; \backslash beta\}\{1\; -\; \backslash beta\backslash cos\backslash theta\}$$

Special relativity / collisions and available energy

Centre-of-mass energy of a fixed-target collision

$$\sqrt{s}=\sqrt{m_1^2{c}^4+m_2^2{c}^4+2E_1{m}_2{c}^2}\backslash sqrt\{s\}\; =\; \backslash sqrt\{m\_1^2\; c^4\; +\; m\_2^2\; c^4\; +\; 2\; E\_1\; m\_2\; c^2\}$$

Special relativity / energy-momentum relation

Relativistic energy from momentum

$$E^2=(pc{)}^2+(m{c}^2{)}^{2}E^2\; =\; (pc)^2\; +\; (mc^2)^2$$

Special relativity / four-vectors and invariant mass

Invariant mass of two particles

$$M{c}^2=\sqrt{(E_1+E_2{)}^2-\mathrm{\mid }{\vec{p}}_1+{\vec{p}}_2{\mathrm{\mid }}^2{c}^2}M\; c^2\; =\; \backslash sqrt\{(E\_1\; +\; E\_2)^2\; -\; \backslash vert\; \backslash vec\; p\_1\; +\; \backslash vec\; p\_2\backslash vert\; ^2\; c^2\}$$

Special relativity / length contraction

Length contraction

$$L=\frac{L_0}{\gamma }=L_0\sqrt{1-v^2\mathrm{/}{c}^2}L\; =\; \backslash frac\{L\_0\}\{\backslash gamma\}\; =\; L\_0\backslash sqrt\{1\; -\; v^2/c^2\}$$

Special relativity / relativistic Doppler effect

Relativistic Doppler shift

$$f_{obs}=f\sqrt{\frac{1\pm \beta }{1\mp \beta }}f\_\{obs\}\; =\; f\backslash sqrt\{\backslash frac\{1\; \backslash pm\; \backslash beta\}\{1\; \backslash mp\; \backslash beta\}\}$$

Special relativity / relativistic kinetic energy

Speed from relativistic kinetic energy

$$\gamma =1+\frac{K}{m{c}^2},v=c\sqrt{1-1\mathrm{/}{\gamma }^2}\backslash gamma\; =\; 1\; +\; \backslash frac\{K\}\{mc^2\},\backslash quad\; v\; =\; c\backslash sqrt\{1\; -\; 1/\backslash gamma^2\}$$

Special relativity / relativistic momentum and energy

Relativistic momentum and energy

$$p=\gamma mv,E=\gamma m{c}^2,K=(\gamma -1)m{c}^{2}p\; =\; \backslash gamma\; m\; v,\backslash quad\; E\; =\; \backslash gamma\; m\; c^2,\backslash quad\; K\; =\; (\backslash gamma\; -\; 1)\; m\; c^2$$

Special relativity / rest energy

Rest energy (mass-energy equivalence)

$$E_0=m{c}^{2}E\_0\; =\; m\; c^2$$

Special relativity / spacetime interval

Spacetime interval between two events

$$s^2=(c\mathrm{\Delta }t{)}^2-\mathrm{\Delta }{x}^{2}s^2\; =\; (c\backslash ,\backslash Delta\; t)^2\; -\; \backslash Delta\; x^2$$

Special relativity / time dilation

Time dilation

$$\mathrm{\Delta }t=\gamma \mathrm{\Delta }{t}_0=\frac{\mathrm{\Delta }{t}_0}{\sqrt{1-v^2\mathrm{/}{c}^2}}\backslash Delta\; t\; =\; \backslash gamma\backslash ,\backslash Delta\; t\_0\; =\; \backslash frac\{\backslash Delta\; t\_0\}\{\backslash sqrt\{1\; -\; v^2/c^2\}\}$$

Special relativity / velocity addition

Relativistic velocity addition

$$u=\frac{u^{\mathrm{\prime }}+v}{1+u^{\mathrm{\prime }}v\mathrm{/}{c}^2}u\; =\; \backslash frac\{u\text{'}\; +\; v\}\{1\; +\; u\text{'}v/c^2\}$$

Spin and qubits / Rabi oscillations

Rabi oscillation of a driven two-level system

$$P_e(t)=\frac{{\mathrm{\Omega }}^2}{{\mathrm{\Omega }}^2+{\mathrm{\Delta }}^2}{\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\sqrt{{\mathrm{\Omega }}^2+{\mathrm{\Delta }}^2}t}{2}\right)P\_e(t)\; =\; \backslash frac\{\backslash Omega^2\}\{\backslash Omega^2\; +\; \backslash Delta^2\}\backslash sin^2\backslash !\backslash left(\backslash frac\{\backslash sqrt\{\backslash Omega^2\; +\; \backslash Delta^2\}\backslash ,t\}\{2\}\backslash right)$$

Spin and qubits / distinguishing quantum states

Fidelity and trace distance of two qubit states

$$F=\frac{1}{2}[1+\vec{r}\cdot \vec{s}+\sqrt{(1-\mathrm{\mid }\vec{r}{\mathrm{\mid }}^2)(1-\mathrm{\mid }\vec{s}{\mathrm{\mid }}^2)}]F\; =\; \backslash tfrac12\backslash left[1\; +\; \backslash vec\; r\backslash cdot\backslash vec\; s\; +\; \backslash sqrt\{(1\; -\; \backslash vert\; \backslash vec\; r\backslash vert\; ^2)(1\; -\; \backslash vert\; \backslash vec\; s\backslash vert\; ^2)\}\backslash right]$$

Spin and qubits / entanglement and Bell inequalities

CHSH Bell-test value for the singlet state

$$S=E(a,b)-E(a,b^{\mathrm{\prime }})+E(a^{\mathrm{\prime }},b)+E(a^{\mathrm{\prime }},b^{\mathrm{\prime }}),E(x,y)=-\cos (x-y)S\; =\; E(a,b)\; -\; E(a,b\text{'})\; +\; E(a\text{'},b)\; +\; E(a\text{'},b\text{'}),\backslash quad\; E(x,y)\; =\; -\backslash cos(x\; -\; y)$$

Spin and qubits / mixed states and density matrices

Purity and entropy of a qubit state

$$\rho =\frac{1}{2}(I+\vec{r}\cdot \vec{\sigma }),S=-\sum _i{\lambda }_i{\log }_2{\lambda }_i\backslash rho\; =\; \backslash tfrac12(I\; +\; \backslash vec\; r\backslash cdot\backslash vec\backslash sigma),\backslash quad\; S\; =\; -\backslash sum\_i\; \backslash lambda\_i\backslash log\_2\backslash lambda\_i$$

Spin and qubits / qubit states

Qubit state on the Bloch sphere

$$\mathrm{\mid }\psi ⟩=\cos \frac{\theta }{2}\mathrm{\mid }0⟩+e^{i\varphi }\sin \frac{\theta }{2}\mathrm{\mid }1⟩\backslash vert\; \backslash psi\backslash rangle\; =\; \backslash cos\backslash tfrac\{\backslash theta\}\{2\}\backslash vert\; 0\backslash rangle\; +\; e^\{i\backslash phi\}\backslash sin\backslash tfrac\{\backslash theta\}\{2\}\backslash vert\; 1\backslash rangle$$

Spin and qubits / rotations of angular momentum states

Spin rotation amplitude (Wigner small-d matrix)

$$d_{m^{\mathrm{\prime }}m}^j(\beta )=⟨j{m}^{\mathrm{\prime }}\mathrm{\mid }{e}^{-i\beta J_y\mathrm{/}\mathrm{\hslash }}\mathrm{\mid }jm⟩d^j\_\{m\text{'}m\}(\backslash beta)\; =\; \backslash langle\; j\; m\text{'}\backslash vert\; e^\{-i\backslash beta\; J\_y/\backslash hbar\}\backslash vert\; j\; m\backslash rangle$$

Spin and qubits / spin measurement

Spin-1/2 measurement along an axis

$$P_{\uparrow }=\mathrm{\mid }⟨{+}_n\mathrm{\mid }\psi ⟩{\mathrm{\mid }}^2=\frac{1}{2}(1+\widehat{r}\cdot \widehat{n})P\_\backslash uparrow\; =\; \backslash vert\; \backslash langle\; +\_n\backslash vert\; \backslash psi\backslash rangle\backslash vert\; ^2\; =\; \backslash tfrac12\backslash left(1\; +\; \backslash hat\; r\backslash cdot\backslash hat\; n\backslash right)$$

Spin and qubits / spin precession

Larmor precession of a spin-1/2

$${\omega }_L=\mathrm{\mid }\gamma \mathrm{\mid }B,P_{+x}(t)={\cos }^2\text{\hspace{0.17em}⁣}\left(\frac{{\omega }_{L}t}{2}\right)\backslash omega\_L\; =\; \backslash vert\; \backslash gamma\backslash vert\; B,\backslash quad\; P\_\{+x\}(t)\; =\; \backslash cos^2\backslash !\backslash left(\backslash tfrac\{\backslash omega\_L\; t\}\{2\}\backslash right)$$

Statistical physics / Bose-Einstein condensation

Bose-Einstein condensation temperature

$$T_c=\frac{2\pi {\mathrm{\hslash }}^2}{m{k}_B}{\left(\frac{n}{\zeta (3\mathrm{/}2)}\right)}^{2\mathrm{/}3}T\_c\; =\; \backslash frac\{2\backslash pi\backslash hbar^2\}\{m\; k\_B\}\backslash left(\backslash frac\{n\}\{\backslash zeta(3/2)\}\backslash right)^\{2/3\}$$

Statistical physics / Einstein model of a solid

Quantum oscillator thermodynamics (Einstein model)

$$C_V=3R{\left(\frac{{\mathrm{\Theta }}_E}{T}\right)}^2\frac{e^{{\mathrm{\Theta }}_E\mathrm{/}T}}{{(e^{{\mathrm{\Theta }}_E\mathrm{/}T}-1)}^2}C\_V\; =\; 3R\backslash left(\backslash frac\{\backslash Theta\_E\}\{T\}\backslash right)^2\backslash frac\{e^\{\backslash Theta\_E/T\}\}\{\backslash left(e^\{\backslash Theta\_E/T\}\; -\; 1\backslash right)^2\}$$

Statistical physics / blackbody radiation

Fraction of blackbody radiation in a wavelength band

$$F_{0\to \lambda }=\frac{15}{{\pi }^4}{\int }_{hc\mathrm{/}\lambda k_{B}T}^{\mathrm{\infty }}\frac{t^3}{e^t-1}dtF\_\{0\backslash to\backslash lambda\}\; =\; \backslash frac\{15\}\{\backslash pi^4\}\backslash int\_\{hc/\backslash lambda\; k\_BT\}^\{\backslash infty\}\backslash frac\{t^3\}\{e^t\; -\; 1\}\backslash ,dt$$

Peak of blackbody radiation (Wien's displacement law)

$${\lambda }_{max}T=b=\frac{hc}{4.965k_B}\backslash lambda\_\{max\}\; T\; =\; b\; =\; \backslash frac\{hc\}\{4.965\backslash ,k\_B\}$$

Radiated power (Stefan-Boltzmann law)

$$P=\epsilon \sigma A{T}^4,\sigma =\frac{2{\pi }^5{k}_B^4}{15h^3{c}^2}P\; =\; \backslash varepsilon\backslash sigma\; A\; T^4,\backslash quad\; \backslash sigma\; =\; \backslash frac\{2\backslash pi^5\; k\_B^4\}\{15\; h^3\; c^2\}$$

Spectral radiance of a blackbody (Planck's law)

$$B_{\lambda }(T)=\frac{2h{c}^2}{{\lambda }^5}\frac{1}{e^{hc\mathrm{/}(\lambda k_{B}T)}-1}B\_\backslash lambda(T)\; =\; \backslash frac\{2hc^2\}\{\backslash lambda^5\}\backslash ,\backslash frac\{1\}\{e^\{hc/(\backslash lambda\; k\_B\; T)\}\; -\; 1\}$$

Statistical physics / free-electron model of metals

Fermi energy of a free-electron gas

$$E_F=\frac{{\mathrm{\hslash }}^2}{2m_e}{\left(3{\pi }^{2}n\right)}^{2\mathrm{/}3}E\_F\; =\; \backslash frac\{\backslash hbar^2\}\{2m\_e\}\backslash left(3\backslash pi^2\; n\backslash right)^\{2/3\}$$

Fermi energy of a metal from its density

$$n=\frac{z\rho N_A}{M},E_F=\frac{{\mathrm{\hslash }}^2}{2m_e}{\left(3{\pi }^{2}n\right)}^{2\mathrm{/}3}n\; =\; \backslash frac\{z\backslash rho\; N\_A\}\{M\},\backslash quad\; E\_F\; =\; \backslash frac\{\backslash hbar^2\}\{2m\_e\}\backslash left(3\backslash pi^2\; n\backslash right)^\{2/3\}$$

Statistical physics / partition functions and Boltzmann populations

Thermodynamics of a two-level system

$$\frac{N_1}{N_0}=\frac{g_1}{g_0}{e}^{-\mathrm{\Delta }E\mathrm{/}{k}_{B}T},\frac{C}{k_B}=x^2{p}_0{p}_1\backslash frac\{N\_1\}\{N\_0\}\; =\; \backslash frac\{g\_1\}\{g\_0\}e^\{-\backslash Delta\; E/k\_BT\},\backslash quad\; \backslash frac\{C\}\{k\_B\}\; =\; x^2\; p\_0\; p\_1$$

Statistical physics / phonons and the Debye model

Debye heat capacity of a solid

$$C_V=9R{\left(\frac{T}{{\mathrm{\Theta }}_D}\right)}^3{\int }_0^{{\mathrm{\Theta }}_D\mathrm{/}T}\frac{x^4{e}^x}{(e^x-1{)}^2}dxC\_V\; =\; 9R\backslash left(\backslash frac\{T\}\{\backslash Theta\_D\}\backslash right)^3\backslash int\_0^\{\backslash Theta\_D/T\}\backslash frac\{x^4\; e^x\}\{(e^x\; -\; 1)^2\}\backslash ,dx$$

Statistical physics / photon gas

Energy and photon density of blackbody radiation

$$u=\frac{{\pi }^2(k_{B}T{)}^4}{15(\mathrm{\hslash }c{)}^3},n=\frac{2\zeta (3)}{{\pi }^2}{\left(\frac{k_{B}T}{\mathrm{\hslash }c}\right)}^{3}u\; =\; \backslash frac\{\backslash pi^2\; (k\_BT)^4\}\{15(\backslash hbar\; c)^3\},\backslash quad\; n\; =\; \backslash frac\{2\backslash zeta(3)\}\{\backslash pi^2\}\backslash left(\backslash frac\{k\_BT\}\{\backslash hbar\; c\}\backslash right)^3$$

Statistical physics / quantum concentration

Thermal de Broglie wavelength

$${\lambda }_{th}=\frac{h}{\sqrt{2\pi m{k}_{B}T}}\backslash lambda\_\{th\}\; =\; \backslash frac\{h\}\{\backslash sqrt\{2\backslash pi\; m\; k\_B\; T\}\}$$

Statistical physics / quantum statistics

Bose-Einstein occupation of a mode

$$\bar{n}(E)=\frac{1}{e^{(E-\mu )\mathrm{/}{k}_{B}T}-1}\backslash bar\; n(E)\; =\; \backslash frac\{1\}\{e^\{(E\; -\; \backslash mu)/k\_BT\}\; -\; 1\}$$

Fermi-Dirac occupation of a state

$$f(E)=\frac{1}{e^{(E-\mu )\mathrm{/}{k}_{B}T}+1}f(E)\; =\; \backslash frac\{1\}\{e^\{(E\; -\; \backslash mu)/k\_BT\}\; +\; 1\}$$

Statistical physics / quantum statistics of gases

Chemical potential of an ideal Fermi gas

$$n=\frac{2}{{\lambda }_{th}^3}{f}_{3\mathrm{/}2}\text{\hspace{0.17em}⁣}\left(e^{\mu \mathrm{/}{k}_{B}T}\right),f_{3\mathrm{/}2}(z)=-{\mathrm{L}\mathrm{i}}_{3\mathrm{/}2}(-z)n\; =\; \backslash frac\{2\}\{\backslash lambda\_\{th\}^3\}\backslash ,f\_\{3/2\}\backslash !\backslash left(e^\{\backslash mu/k\_BT\}\backslash right),\backslash quad\; f\_\{3/2\}(z)\; =\; -\backslash mathrm\{Li\}\_\{3/2\}(-z)$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified quantum physics capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (special relativity, photons and matter waves, the hydrogen atom and angular momentum, quantum wells and tunnelling, spin, qubits and quantum gates, nuclear and particle physics, statistical physics); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{quantum_physics_ai999x_1_0_0,
  title  = {quantum-physics-ai999x: an offline, verified quantum physics solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/quantum-physics-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/quantum-physics-ai999x/discussions or visit https://ai999x.com.

Support the project

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Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com