Special relativity, photons and matter waves, the hydrogen atom, quantum wells and tunnelling, spin and qubits, quantum gates, nuclear and particle physics and statistical physics.
Quantum Physics (quantum-physics-ai999x) is an offline Python package for quantum physics. It covers special relativity, photons and matter waves, the hydrogen atom, quantum wells and tunnelling, spin and qubits, quantum gates, nuclear and particle physics and statistical physics.
It has 75 capabilities — 75 computational procedures — in 11 topics, the largest being Statistical physics, Special relativity, Quantum wells, Spin and qubits, Hydrogen atom and Photons and matter waves. Ask a question in words with ask(), or call a capability by name with typed inputs with solve(). Every answer carries its value and unit, the steps that produced it and the formula in LaTeX.
When a question is not determined — a value is missing, an input is out of range, or the request is outside the package — it says so and names what it needs, instead of guessing. It runs on an ordinary CPU without a network connection (Python >=3.10), and the same question always gets the same answer.
pip install quantum-physics-ai999x hf download AI999X/quantum-physics-ai999x --local-dir quantum-physics-ai999x
Each question below was put to quantum-physics-ai999x and the answer shown is the one it returned, with the steps it took.
Lorentz factor: v = 0.8 c
v = 239833966.4 m/sgamma = 1.666666667, beta = 0.8, gamma_minus_1 = 0.6666666667Hydrogen transition: Z = 1, n_upper = 3, n_lower = 2
Z = 1, n_upper = 3, n_lower = 2E_photon_eV = 1.8896796 eV, lam = 6.561122764e-07 m, lambda_nm = 656.1122764 nm, f = 4.569224945e+14 Hz, series = Balmer nmPhotoelectric effect: lambda = 250 nm, W = 4.3 eV
lam = 250 nm, W = 4.3 eVE_photon_eV = 4.959367937 eV, K_max_eV = 0.6593679373 eV, V_stop = 0.6593679373 V, lambda_threshold = 2.883353452e-07 m, lambda_threshold_nm = 288.3353452 nm, f_threshold = 1.039735374e+15 Hzm.solve("two_qubit_circuit", start="00", gates="H0 CNOT01") start = 00, gates = H0 CNOT01p00 = 0.5, p01 = 0, p10 = 0, p11 = 0.5, concurrence = 1, p_q0_is_1 = 0.5, p_q1_is_1 = 0.5Barrier transmission: E = 1 eV, V0 = 2 eV, a = 0.5 nm, m = 9.109e-31 kg
E = 1 eV, V0 = 2 eV, a = 0.5 nm, m = 9.109e-31 kgT = 0.02354963067, R = 0.9764503693, kappa_a = 2.56152966, regime = tunnelling (E < V0)Clebsch-Gordan coefficient: j1 = 0.5, m1 = 0.5, j2 = 0.5, m2 = -0.5, J = 1, M = 0
j1 = 0.5, m1 = 0.5, j2 = 0.5, m2 = -0.5, J = 1, M = 0cg = 0.7071067812, cg_squared = 0.5, cg_squared_exact = 1/2Semi-empirical mass formula: Z = 26, A = 56
Z = 26, A = 56B_MeV = 495.3836999 MeV, B_per_A_MeV = 8.846137499 MeV, mass_u = 55.93158207 MeV, volume_MeV = 882 MeV, surface_MeV = 260.5425666 MeV, coulomb_MeV = 120.7962953 MeV, asymmetry_MeV = 6.771428571 MeV, pairing_MeV = 1.493990342 MeV, Z_stable = 25.27970457 MeVCoulomb barrier: Z1 = 1, A1 = 1, Z2 = 1, A2 = 1, r0 = 1 fm
Z1 = 1, A1 = 1, Z2 = 1, A2 = 1, r0 = 1 fmV_MeV = 0.7199822734 MeV, R = 2e-15 m, R_fm = 2 fmEvery snippet was run as shown; the output under it is what it printed.
import quantum_physics_ai999x as m
a = m.ask("Lorentz factor: v = 0.8 c")
print(a.outcome)
print(a.values)
for step in a.steps:
print(step["name"])SOLVED
{'gamma': 1.66666666667, 'beta': 0.8, 'gamma_minus_1': 0.666666666667}
Lorentz factor
import quantum_physics_ai999x as m
a = m.solve("two_qubit_circuit", start="00", gates="H0 CNOT01")
print(a.outcome)
print(a.values)SOLVED
{'p00': 0.5, 'p01': 0.0, 'p10': 0.0, 'p11': 0.5, 'concurrence': 1.0, 'p_q0_is_1': 0.5, 'p_q1_is_1': 0.5}
import quantum_physics_ai999x as m
# an input is missing: the answer says what it needs instead of guessing
a = m.solve("two_qubit_circuit")
print(a.outcome)
print(a.message)NEEDS_INPUT
two_qubit_circuit needs start, gates (initial basis state |q0 q1>; gates left to right: H0, X1, Rz0(t), ..., CNOT01 (control q0), CNOT10, CZ, SWAP). Example: solve('two_qubit_circuit', start='00', gates='H0 CNOT01').
quantum-physics-ai999x "Lorentz factor: v = 0.8 c"
gamma = 1.66667, beta = 0.8, gamma_minus_1 = 0.666667 1. Lorentz factor: gamma = 1/sqrt((1 - beta)(1 + beta)), beta = v/c with v = 2.39834e+08 m/s -> gamma = 1.66667, beta = 0.8, gamma_minus_1 = 0.666667 Assumptions: 0 <= v < c …
Every value in these figures was computed by the package, each point a run of solve(); in a notebook, plot() and plot_relation() draw a capability's curves the same way.
12 of its 75 formulas; every one is in the capability browser below.
Search by name, formula, input or topic. Each capability opens to show what it takes, what it returns, when it applies, and a question it answered.
clebsch_gordanThe coefficient <j1 m1; j2 m2 | J M> for coupling two angular momenta (Condon-Shortley convention), with its exact square.
j1 first angular momentum (whole or half-whole)m1 its projectionj2 second angular momentumm2 its projectionJ total angular momentumM total projectioncg Clebsch-Gordan coefficientcg_squared its square (a probability)cg_squared_exact the square as an exact fractionwigner_3j_symbolThe Wigner 3j symbol (j1 j2 J; m1 m2 M), nonzero only when m1 + m2 + M = 0.
j1 first angular momentum (whole or half-whole)m1 its projectionj2 second angular momentumm2 its projectionJ total angular momentumM total projectionw3j the 3j symbolw3j_squared its squarebohr_energy_levelEnergy, ionization energy, orbit radius and orbital speed of level n of a one-electron atom or ion with nuclear charge Z.
Z atomic number (nuclear charge; 1 for hydrogen, 2 for He+ ...) integern principal quantum number integerE_eV energy of the level (negative: bound) [eV]E the same in J [J]ionization_eV energy needed to ionize from this level [eV]r Bohr orbit radius [m]r_nm orbit radius in nm [nm]v orbital speed [m/s]hydrogen_transition_reduced_massPhoton energy and wavelength of a transition of a one-electron atom whose nucleus has mass M (in u): R_M = R_inf mu/m_e.
Z atomic number (1 for hydrogen) integern_upper upper level integern_lower lower level integerM_u mass of the nucleus in u (1.007276 for the proton, 2.013553 for the deuteron)E_photon_eV photon energy [eV]lam vacuum wavelength [m]lambda_nm vacuum wavelength in nm [nm]f frequency [Hz]series spectral series (Lyman, Balmer, ...)mu_ratio mu/m_emoseley_k_alphaEnergy and wavelength of the K-alpha X-ray line of element Z, with one unit of screening (Z - 1).
Z atomic number of the target element integerE_keV photon energy [keV]lam wavelength [m]lambda_nm wavelength in nm [nm]lambda_pm wavelength in pm [pm]f frequency [Hz]lande_g_zeemanThe Landé g factor of a level L, S, J (with g_s = 2) and the energy shift g_J mu_B B mJ in a weak magnetic field.
L orbital angular momentum quantum numberS spin quantum numberJ total angular momentum quantum numbermJ magnetic quantum numberB magnetic field (T) [T]g_J Landé g factordE Zeeman energy shift [J]dE_eV the shift in eV [eV]g_J_exact g_J as an exact fractionhydrogen_transitionEnergy, wavelength and frequency of the photon emitted when a one-electron atom drops from n_upper to n_lower.
Z atomic number (1 for hydrogen) integern_upper upper level integern_lower lower level integerE_photon_eV photon energy [eV]lam vacuum wavelength [m]lambda_nm vacuum wavelength in nm [nm]f frequency [Hz]series spectral series (Lyman, Balmer, ...)hydrogen_radial_probabilityThe probability that the electron of the hydrogen-like state n, l lies within distance R of the nucleus.
n principal quantum number integerl orbital quantum number (0 <= l < n) integerZ atomic number integerR radius (pm, nm, m) [m]P_inside probability within RP_outside probability beyond RR_a0 R in units of a0hydrogen_radial_expectations<r>, <r^2>, <1/r> and the radial spread of the state n, l of a one-electron atom with nuclear charge Z (in Bohr radii).
n principal quantum number integerl orbital quantum number (0 <= l < n) integerZ atomic number integerr_mean_a0 <r> in units of a0r2_mean_a0sq <r^2> in units of a0^2inv_r_mean_per_a0 <1/r> in units of 1/a0r_mean <r> in m [m]dr_a0 radial spread sqrt(<r^2> - <r>^2) in a0two_body_decayQ value, momentum and kinetic energies of the two products when a particle or nucleus at rest decays into two bodies (alpha decay, pi -> mu nu, ...).
M mass of the parentm1 mass of product 1m2 mass of product 2 (0 for a photon or neutrino)unit unit of the masses: MeV/c^2 (default) or u choice optionalQ_MeV Q value (M - m1 - m2) c^2 [MeV]p_MeV momentum of each product times c [MeV]E1_MeV total energy of product 1 [MeV]E2_MeV total energy of product 2 [MeV]K1_MeV kinetic energy of product 1 [MeV]K2_MeV kinetic energy of product 2 [MeV]semi_empirical_mass_formulaThe liquid-drop (Bethe-Weizsaecker) binding energy of a nucleus with Z protons and mass number A, its terms, the binding energy per nucleon, the atomic mass and the most stable Z for that A.
Z number of protons integerA mass number integerB_MeV binding energy [MeV]B_per_A_MeV binding energy per nucleon [MeV]mass_u atomic mass in u (Z m(1H) + N m_n - B/c^2)volume_MeV volume term a_V A [MeV]surface_MeV surface term a_S A^(2/3) [MeV]coulomb_MeV Coulomb term a_C Z(Z - 1)/A^(1/3) [MeV]asymmetry_MeV asymmetry term a_A (A - 2Z)^2/A [MeV]pairing_MeV pairing term (+ even-even, 0 odd A, - odd-odd) [MeV]Z_stable most stable (continuous) Z for this A, where dB/dZ = 0radioactive_dating_ageThe time needed for a radioactive sample to decay to a given fraction of its original amount (for example carbon-14 dating).
fraction fraction of the original nuclei (or activity) remaining, 0 < f <= 1 (or a percentage)T_half half-life [s]t elapsed time (age) [s]half_lives number of half-livesdecay_constant decay constant [1/s]coulomb_barrierThe Coulomb potential energy of two nuclei just touching, with radii R = r0 A^(1/3).
Z1 charge number of nucleus 1 integerA1 mass number of nucleus 1 integerZ2 charge number of nucleus 2 integerA2 mass number of nucleus 2 integerr0 radius parameter (typically 1.2 fm) [m]V_MeV Coulomb barrier height [MeV]R touching distance r0 (A1^(1/3) + A2^(1/3)) [m]R_fm the same in fm [fm]nuclide_decayThe fraction of a named radionuclide (C-14, Co-60, I-131, U-238, ...) left after time t, with its tabulated half-life, decay constant and mean life.
nuclide the radionuclide (e.g. C-14, Cs-137, I-131) choicet elapsed time [s]fraction fraction of the parent nuclide remaininghalf_life half-life (ICRP-107) [s]half_life_years half-life in Julian yearsdecay_constant decay constant [1/s]mean_life mean life [s]specific_activity_per_atom activity per original atom at time t (Bq) [1/s]decay_chain_parent_daughterNumbers and activities of a parent and its radioactive daughter after time t, starting from pure parent (Bateman solution).
T_half_P half-life of the parent [s]T_half_D half-life of the daughter [s]N0 initial number of parent nucleit elapsed time [s]N_parent parent nucleiN_daughter daughter nucleiA_parent parent activity (Bq) [1/s]A_daughter daughter activity (Bq) [1/s]activity_ratio A_daughter/A_parent (tends to 1 in secular equilibrium)nuclear_reaction_thresholdQ value of a + X -> Y + b from the masses (in u) and, for an endothermic reaction, the minimum beam kinetic energy on a target at rest (relativistic).
m_a projectile mass (u)m_X target mass (u)m_Y heavy product mass (u)m_b light product mass (u)Q_MeV Q value [MeV]K_threshold_MeV threshold kinetic energy of the projectile (0 if exothermic) [MeV]endothermic whether the reaction needs energy (Q < 0)radioactive_decay_lawNuclei remaining, fraction, decay constant, mean life and activity after time t, from the half-life and the initial number of nuclei.
T_half half-life [s]N0 initial number of nucleit elapsed time [s]N nuclei remainingfraction fraction remaining N/N0decay_constant decay constant ln2/T_half [1/s]mean_life mean life 1/lambda [s]A0 initial activity (Bq) [1/s]A activity at time t (Bq) [1/s]half_lives number of half-lives elapsedparticle_decay_lengthLorentz factor, lab-frame lifetime and mean decay length beta gamma c tau of a pion, kaon, muon, hyperon, charm or bottom meson with momentum p.
particle the particle (pi+, K+, K_S0, K_L0, mu-, tau-, n, Lambda, ...) choicep momentum in GeV/cgamma Lorentz factorbeta v/cbeta_gamma beta gamma = p/(m c)lifetime_lab mean lifetime in the lab [s]mean_decay_length mean decay length beta gamma c tau [m]proper_lifetime mean proper lifetime (PDG) [s]mass_MeV mass (MeV/c^2, PDG)compton_scatteringWavelength and energy of a photon scattered by a free electron at rest through angle theta, and the electron's kinetic energy.
lam incident wavelength (pm, nm, m) [m]theta scattering angle of the photon (deg or rad) [rad]lambda_prime scattered wavelength [m]lambda_prime_pm scattered wavelength in pm [pm]shift_pm Compton shift lambda' - lambda in pm [pm]E_in_keV incident photon energy [keV]E_out_keV scattered photon energy [keV]K_electron_keV kinetic energy of the recoil electron [keV]de_broglie_wavelength_mvThe de Broglie wavelength lambda = h/(m v) of a particle of mass m moving at speed v, and the relativistic h/(gamma m v).
m mass of the particle [kg]v speed (may be written as a fraction of c) [m/s]p momentum m v [kg*m/s]lam de Broglie wavelength h/(m v) [m]lambda_nm the same in nm [nm]lambda_rel_nm relativistic wavelength h/(gamma m v) in nm [nm]inv_lambda 1/lambda = m v/h in 1/m (well scaled for heavy bodies, whose lambda is tiny)beta v/cde_broglie_acceleratedThe de Broglie wavelength of an electron or proton accelerated from rest through a voltage V, non-relativistic and relativistic.
V accelerating voltage [V]particle the accelerated particle (charge e) choicelambda_nonrel wavelength h/sqrt(2 m e V) [m]lambda_nonrel_nm the same in nm [nm]lambda_rel relativistic wavelength [m]lambda_rel_nm the same in nm [nm]p relativistic momentum [kg*m/s]K_eV kinetic energy [eV]heisenberg_minimum_uncertaintyThe smallest momentum and velocity uncertainty compatible with a position uncertainty dx: dp = hbar/(2 dx).
dx position uncertainty (standard deviation) [m]m mass of the particle [kg]dp minimum momentum uncertainty [kg*m/s]dp_eV_per_c the same in eV/cdv minimum velocity uncertainty [m/s]photoelectric_effectMaximum kinetic energy of photoelectrons, stopping potential and threshold wavelength from the light's wavelength and the metal's work function.
lam wavelength of the light (nm, m ...) [m]W work function of the metal (eV, J) [J]E_photon_eV photon energy [eV]K_max_eV maximum kinetic energy of the photoelectrons [eV]V_stop stopping potential [V]lambda_threshold threshold (cut-off) wavelength [m]lambda_threshold_nm threshold wavelength in nm [nm]f_threshold threshold frequency [Hz]photon_energyThe energy E = h c/lambda, frequency and momentum h/lambda of a photon of vacuum wavelength lambda.
lam vacuum wavelength (nm, m ...) [m]E photon energy in J [J]E_eV photon energy in eV [eV]f frequency c/lambda [Hz]p photon momentum h/lambda [kg*m/s]coherent_state_photon_statisticsThe probability of n photons in a coherent (laser) state |alpha>, its mean photon number and variance (Poisson statistics).
alpha coherent amplitude |alpha|n photon number integerP_n probability of exactly n photonsmean_n mean photon number |alpha|^2variance photon-number variance |alpha|^2P_0 probability of no photonsgate_rotation_axisThe single rotation (axis n and angle) of the Bloch sphere that a sequence of single-qubit gates performs (global phase ignored).
gates gates applied left to right, e.g. 'H T Rz(pi/2) X' textangle rotation angle in [0, pi] (rad) [rad]angle_deg rotation angle in degreesnx axis xny axis ynz axis zsingle_qubit_circuitThe amplitudes, measurement probabilities and Bloch vector of a qubit after gates such as H, X, S, T and rotations act on |0>, |1>, |+>, ...
start initial state |0>, |1>, |+>, |->, |+i> or |-i> choicegates gates applied left to right, e.g. 'H T Rz(pi/2) X' textalpha_re Re alpha (global phase fixed: first non-zero amplitude real positive)alpha_im Im alphabeta_re Re betabeta_im Im betap0 probability of measuring 0p1 probability of measuring 1bx Bloch xby Bloch ybz Bloch zgrover_searchThe probability of measuring a marked item after k Grover iterations on n qubits with M marked items, and the best number of iterations.
n_qubits number of qubits (search space N = 2^n) integerM number of marked items integerk number of Grover iterations integerP_success probability of a marked item after k iterationsk_opt best iteration count floor(pi/(4 theta))P_at_k_opt success probability at k_opttheta rotation half-angle asin(sqrt(M/N)) [rad]two_qubit_circuitMeasurement probabilities of |00>, |01>, |10>, |11> and the concurrence (entanglement) after a two-qubit gate sequence such as H0 CNOT01 (a Bell state).
start initial basis state |q0 q1> choicegates gates left to right: H0, X1, Rz0(t), ..., CNOT01 (control q0), CNOT10, CZ, SWAP textp00 probability of |00>p01 probability of |01>p10 probability of |10>p11 probability of |11>concurrence concurrence 2|a00 a11 - a01 a10| (0 product, 1 maximally entangled)p_q0_is_1 probability that q0 reads 1p_q1_is_1 probability that q1 reads 1finite_square_wellThe number and energies of the bound states of a particle of mass m in a square well of depth V0 and width a.
V0 depth of the well (eV, J) [J]a width of the well [m]m particle mass [kg]n_bound number of bound statesE1_eV ground-state energy above the bottom of the well [eV]energies_eV all bound-state energies above the bottom (eV), ascendingz0 well strength (a/2) sqrt(2 m V0)/hbarinfinite_well_levelsEnergy E_n = n^2 h^2/(8 m L^2), momentum and de Broglie wavelength of level n of a particle in a one-dimensional box of width L.
m particle mass [kg]L width of the box [m]n quantum number (1, 2, 3, ...) integerE energy of level n [J]E_eV the same in eV [eV]E1_eV ground-state energy [eV]p magnitude of the momentum n pi hbar/L [kg*m/s]wavelength de Broglie wavelength 2L/n [m]infinite_well_transitionEnergy, wavelength and frequency of the photon emitted when a particle in a box of width L drops from n_upper to n_lower.
m particle mass [kg]L width of the box [m]n_upper upper level integern_lower lower level integerE_photon_eV photon energy [eV]lam wavelength [m]lambda_nm wavelength in nm [nm]f frequency [Hz]infinite_well_uncertainty<x>, the position spread dx, the momentum spread dp and the product dx dp of level n of a particle in a box of width L.
n quantum number integerL width of the box [m]x_mean <x> (the centre) [m]dx position spread [m]dx_over_L dx/Ldp momentum spread n pi hbar/L [kg*m/s]product_over_hbar dx dp / hbar (at least 1/2)harmonic_oscillator_forbidden_probabilityThe probability that a harmonic oscillator in level n is found in the classically forbidden region |x| > A_n.
n quantum number (0 to 60) integerP_forbidden probability outside the turning pointsP_allowed probability insidexi_turning turning point in units of sqrt(hbar/(m omega))infinite_well_probabilityThe probability that a particle in level n of a box of width L is found between x1 and x2.
n quantum number integerL width of the box [m]x1 left end of the interval (from the wall) [m]x2 right end of the interval [m]P probabilityclassical classical (uniform) probability (x2 - x1)/Lharmonic_oscillator_levelsEnergy (n + 1/2) hbar omega, zero-point energy, level spacing, classical amplitude and position spread of level n.
m mass [kg]omega angular frequency [rad/s]n quantum number (0, 1, 2, ...) integerE energy of level n [J]E_eV the same in eV [eV]E0_eV zero-point energy [eV]spacing_eV level spacing hbar omega [eV]amplitude classical turning point [m]dx position spread sqrt(<x^2>) [m]step_potentialReflection and transmission probabilities of a particle of energy E meeting a potential step of height V0 (V = 0 for x < 0, V0 for x > 0).
E kinetic energy of the incident particle (on the low side) [J]V0 height of the step [J]R reflection probabilityT transmission probability (ratio of probability currents)k2_over_k1 ratio of wavenumbers sqrt((E - V0)/E) (0 when E <= V0)regime partial transmission (E > V0) or total reflectionbarrier_transmissionTransmission and reflection probabilities of a particle of energy E meeting a rectangular barrier of height V0 and width a (below or above the top).
E kinetic energy of the incident particle [J]V0 barrier height [J]a barrier width [m]m particle mass [kg]T transmission probabilityR reflection probabilitykappa_a kappa a = a sqrt(2m(V0 - E))/hbar (0 above the barrier)regime tunnelling (E < V0) or over the barrierwkb_field_emissionThe WKB tunnelling exponent and probability of an electron at the Fermi level through the triangular barrier W - eFx of a metal in a field F.
W barrier height above the electron's energy (work function, eV) [J]F applied electric field (V/m) [N/C]exponent WKB exponent 2 integral kappa dxT tunnelling probability exp(-exponent)log10_T log10 of the probabilitywidth barrier width W/(eF) [m]width_nm width in nm [nm]fixed_target_cm_energysqrt(s) for a beam particle of mass m1 and total energy E1 striking a target of mass m2 at rest (natural units, GeV).
m1 beam particle mass (GeV/c^2)E1 beam particle total energy (GeV)m2 target mass (GeV/c^2)sqrt_s centre-of-mass energy sqrt(s) (GeV)available energy available for new particles, sqrt(s) - m1 - m2 (GeV)beta_cm speed of the centre-of-mass frame / cgamma_cm its Lorentz factorinvariant_massThe invariant mass M = sqrt((E1 + E2)^2 - |p1 + p2|^2) of two particles, in natural units (GeV, GeV/c, GeV/c^2).
E1 energy of particle 1 (GeV)p1 momentum of particle 1 (px, py, pz in GeV/c) vectorE2 energy of particle 2 (GeV)p2 momentum of particle 2 (px, py, pz in GeV/c) vectorM invariant mass (GeV/c^2)P magnitude of the total momentum (GeV/c)beta_cm speed of the centre-of-mass frame / cgamma_cm Lorentz factor of the centre-of-mass framelength_contractionThe length L = L0/gamma of a rod moving along its length at speed v, given its proper length L0.
L0 proper length (in the rod's rest frame) [m]v speed of the rod (may be written as a fraction of c) [m/s]L contracted length measured in the frame where the rod moves [m]gamma Lorentz factorshortening L0 - L [m]lorentz_factorThe Lorentz factor gamma = 1/sqrt(1 - v^2/c^2) of a speed v.
v speed (may be written as a fraction of c, e.g. 0.8 c) [m/s]gamma Lorentz factorbeta v/cgamma_minus_1 gamma - 1 (computed without cancellation)relativistic_aberrationThe direction of a light ray seen from a frame moving at speed v along +x, given its direction theta (from +x) in the rest frame.
theta direction of travel of the light in the rest frame, from the +x axis (rad or deg) [rad]v speed of the moving frame along +x (may be a fraction of c) [m/s]theta_prime direction of travel in the moving frame (rad) [rad]theta_prime_deg the same angle in degreesdoppler_factor photon energy ratio E'/Erelativistic_dopplerThe observed frequency of light from a source approaching or receding at speed v (along the line of sight).
f frequency in the source frame [Hz]v relative speed (may be a fraction of c) [m/s]motion whether source and observer approach or recede choicef_obs observed frequency [Hz]ratio f_obs / fz redshift z = f/f_obs - 1 (negative for a blueshift)relativistic_energy_from_momentumTotal and kinetic energy and speed of a particle of mass m with momentum p, from E^2 = (pc)^2 + (mc^2)^2.
m rest mass [kg]p momentum [kg*m/s]E total energy [J]K kinetic energy [J]v speed [m/s]beta v/cgamma Lorentz factorE_MeV total energy in MeV [MeV]K_MeV kinetic energy in MeV [MeV]relativistic_momentum_energyMomentum p = gamma m v, total energy E = gamma m c^2, kinetic energy (gamma - 1) m c^2 and rest energy of a mass m at speed v.
m rest mass [kg]v speed (may be written as a fraction of c) [m/s]p relativistic momentum [kg*m/s]E total energy [J]K kinetic energy [J]E0 rest energy m c^2 [J]gamma Lorentz factorE_MeV total energy in MeV [MeV]K_MeV kinetic energy in MeV [MeV]pc_MeV momentum times c in MeV [MeV]relativistic_velocity_additionThe velocity u = (u' + v)/(1 + u'v/c^2) of an object moving at u' in a frame that moves at v (signed, along x).
u_prime velocity of the object in the moving frame (signed; may be a fraction of c) [m/s]v velocity of the moving frame (signed; may be a fraction of c) [m/s]u velocity of the object in the rest frame [m/s]beta u/crest_energyThe rest energy E = m c^2 of a mass m, in J, MeV and kWh.
m rest mass (kg, g ...) [kg]E rest energy in J [J]E_MeV rest energy in MeV [MeV]E_kWh rest energy in kWhspacetime_intervalThe invariant interval s^2 = (c dt)^2 - dx^2 between two events, its type, and the proper time or proper distance.
dt time between the events (signed) [s]dx distance between the events along x (signed) [m]s2 interval s^2 = (c dt)^2 - dx^2 [m^2]nature timelike, spacelike or lightliketau proper time (timelike; else 0) [s]proper_length proper distance (spacelike; else 0) [m]relativistic_speed_from_kinetic_energyThe speed, Lorentz factor and momentum of a particle of mass m with kinetic energy K.
m rest mass [kg]K kinetic energy (J, eV, MeV ...) [J]v speed [m/s]beta v/cgamma Lorentz factorp momentum [kg*m/s]pc_MeV momentum times c in MeV [MeV]time_dilationThe time t = gamma t0 between two ticks of a clock that moves at speed v, given its proper time t0.
t0 proper time (measured by the moving clock) [s]v speed of the clock (may be written as a fraction of c) [m/s]t dilated time measured in the frame where the clock moves [s]gamma Lorentz factordifference t - t0 [s]chsh_bell_testCorrelations E(a, b) = -cos(a - b) of spin measurements on a singlet pair and the CHSH combination S = E(a,b) - E(a,b') + E(a',b) + E(a',b').
a Alice's first setting (angle in the x-z plane) [rad]a2 Alice's second setting a' [rad]b Bob's first setting [rad]b2 Bob's second setting b' [rad]S CHSH valueabs_S |S| (local hidden variables allow at most 2)E_ab E(a, b)E_ab2 E(a, b')E_a2b E(a', b)E_a2b2 E(a', b')violates whether |S| exceeds 2qubit_fidelityThe (Uhlmann) fidelity and the trace distance between two qubit states given by their Bloch vectors r and s.
r Bloch vector of the first state vectors Bloch vector of the second state vectorF fidelity (Tr sqrt(sqrt(rho) sigma sqrt(rho)))^2sqrt_F root fidelitytrace_distance trace distance |r - s|/2larmor_precessionLarmor angular frequency of an electron or proton spin in a field B, and the spin component along x after time t for a spin starting along +x.
particle the spin-1/2 particle choiceB magnetic field along z [T]t time since the spin pointed along +x [s]omega_L Larmor angular frequency |gamma| B [rad/s]f_L Larmor frequency [Hz]Sx_mean <sigma_x> at time t (mean spin along x in units of hbar/2)P_plus_x probability of measuring spin up along +x at time tqubit_entropy_purityPurity Tr(rho^2), von Neumann entropy (bits) and eigenvalues of the qubit density matrix with Bloch vector r.
r Bloch vector (rx, ry, rz), length at most 1 vectorlength |r|purity Tr(rho^2) = (1 + |r|^2)/2S_bits von Neumann entropy in bitslambda_max larger eigenvalue (1 + |r|)/2lambda_min smaller eigenvalue (1 - |r|)/2qubit_state_blochNormalised measurement probabilities, Bloch vector, and Bloch angles of the qubit state alpha|0> + beta|1> (amplitudes need not be normalised).
a_re real part of alphaa_im imaginary part of alphab_re real part of betab_im imaginary part of betap0 probability of 0p1 probability of 1x Bloch x = <sigma_x>y Bloch y = <sigma_y>z Bloch z = <sigma_z>theta polar angle (rad) [rad]phi azimuth in [0, 2 pi) (0 when one amplitude is zero) [rad]norm norm of the given amplitudesrabi_oscillationThe probability of finding a two-level system (qubit) excited after driving it for time t with Rabi frequency Omega and detuning Delta.
Omega Rabi angular frequency [rad/s]Delta detuning (drive minus transition angular frequency) [rad/s]t driving time [s]P_excited excited-state probabilityW generalised Rabi frequency sqrt(Omega^2 + Delta^2) [rad/s]P_max largest excitation probability Omega^2/W^2period oscillation period 2 pi/W [s]spin_rotation_wigner_dThe amplitude d^j_{m'm}(beta) = <j m'| exp(-i beta J_y/hbar) |j m> and the probability of finding m' after rotating |j m> by beta about y.
j angular momentum quantum number (whole or half-whole, at most 20)mp final projection m'm initial projection mbeta rotation angle about the y axis (deg or rad) [rad]d Wigner small-d matrix elementprobability probability |d|^2 of measuring m' along the original axisspin_half_measurementProbabilities of spin up and down along a direction n (angles theta_n, phi_n) for a spin-1/2 state pointing along (theta, phi) on the Bloch sphere.
theta polar angle of the state's Bloch vector [rad]phi azimuth of the state's Bloch vector [rad]theta_n polar angle of the measurement axis [rad]phi_n azimuth of the measurement axis [rad]P_up probability of +hbar/2 along nP_down probability of -hbar/2 along nexpectation <sigma_n> (the mean spin along n in units of hbar/2)bec_critical_temperatureThe critical temperature below which an ideal Bose gas of density n and particle mass m forms a condensate.
n particle density [1/m^3]m particle mass [kg]T_c critical temperature [K]lambda_c thermal wavelength at T_c [m]n_lambda3 phase-space density at T_c, zeta(3/2)bose_einstein_occupationThe mean number of bosons in a single-particle state of energy E at temperature T and chemical potential mu (mu < E).
E energy of the state [J]mu chemical potential (0 for photons and phonons) [J]T temperature [K]n mean occupation 1/(e^x - 1)x (E - mu)/kTclassical_limit Maxwell-Boltzmann value e^(-x)m.solve("bose_einstein_occupation", E=2.870978885078724e-21, mu=0.0, T=300.0)fermi_gas_chemical_potentialThe chemical potential of an ideal spin-1/2 Fermi gas of density n and particle mass m at temperature T, from degenerate to classical.
n particle density [1/m^3]T temperature [K]m particle mass [kg]mu_eV chemical potential [eV]eta mu/kTE_F_eV Fermi energy at T = 0 [eV]mu_over_EF mu/E_FT_over_TF T/T_Fm.solve("fermi_gas_chemical_potential", n=1e+28, T=100000.0, m=9.1093837139e-31)debye_heat_capacityThe molar heat capacity of a crystalline solid in the Debye model at temperature T.
T temperature [K]theta_D Debye temperature [K]C_molar molar heat capacity C_V [J/(mol*K)]C_over_3R C_V / 3R (tends to 1 at high T)debye_integral integral_0^(theta/T) x^4 e^x/(e^x - 1)^2 dxphoton_gasEnergy density, photon number density, mean photon energy and radiation pressure of thermal radiation at temperature T.
T temperature [K]u energy density (J/m^3) [Pa]n photon number density [1/m^3]mean_E_eV mean photon energy [eV]mean_E_over_kT mean photon energy in units of kT (2.701...)pressure radiation pressure u/3 [Pa]fermi_energy_free_electronsFermi energy, temperature, wavenumber and speed of a free-electron gas with electron density n.
n conduction-electron density (per m^3) [1/m^3]E_F_eV Fermi energy [eV]E_F Fermi energy in J [J]T_F Fermi temperature E_F/k [K]k_F Fermi wavenumber (3 pi^2 n)^(1/3) [1/m]v_F Fermi speed [m/s]fermi_energy_metalConduction-electron density and Fermi energy of a metal from its mass density, molar mass and valence (free-electron model).
rho mass density of the metal [kg/m^3]M molar mass (g/mol or kg/mol) [kg/mol]z conduction electrons per atom (valence) integern conduction-electron density [1/m^3]E_F_eV Fermi energy [eV]T_F Fermi temperature [K]v_F Fermi speed [m/s]fermi_dirac_occupationThe probability that a single-particle state of energy E is occupied by a fermion at temperature T and chemical potential mu.
E energy of the state [J]mu chemical potential (Fermi level) [J]T temperature [K]f occupation probability 1/(e^x + 1)one_minus_f probability the state is emptyx (E - mu)/kTblackbody_band_fractionThe fraction of a blackbody's emitted power between two wavelengths at temperature T, and that band's radiant exitance.
lam1 shorter wavelength [m]lam2 longer wavelength [m]T temperature [K]F_band fraction of the total emission between lambda1 and lambda2F_below_lam1 fraction emitted below lambda1F_below_lam2 fraction emitted below lambda2M_band radiant exitance in the band (blackbody) [W/m^2]m.solve("blackbody_band_fraction", lam1=1e-12, lam2=1.0, T=5778.0)wien_peakThe wavelength and frequency at which a blackbody at temperature T radiates most strongly.
T temperature of the blackbody [K]lambda_max peak wavelength (of B_lambda) [m]lambda_max_nm peak wavelength in nm [nm]nu_max peak frequency (of B_nu; not c/lambda_max) [Hz]b Wien's constant b = hc/(x k) [m*K]einstein_oscillatorMean quantum number, mean energy, heat capacity and entropy of a quantum harmonic oscillator, and the molar heat capacity of an Einstein solid.
theta_E Einstein temperature hbar omega/k [K]T temperature [K]n_mean mean number of quanta 1/(e^x - 1)U_eV mean energy including the zero-point energy [eV]C_over_k heat capacity per oscillator in units of kC_molar molar heat capacity of an Einstein solid 3R C/k [J/(mol*K)]S_over_k entropy per oscillator in units of kstefan_boltzmannRadiant exitance and total power emitted by a surface of area A, emissivity e at temperature T.
T surface temperature [K]emissivity emissivity (1 for a blackbody)A radiating area [m^2]sigma Stefan-Boltzmann constant 2 pi^5 k^4/(15 h^3 c^2) [W/(m^2*K^4)]M radiant exitance e sigma T^4 [W/m^2]P total radiated power [W]planck_spectral_radianceThe spectral radiance B_lambda of a blackbody at temperature T and wavelength lambda, and the photon occupation number of that mode.
lam wavelength [m]T temperature [K]B_lambda spectral radiance (W per m^2 per sr per m of wavelength) [W/m^3]B_lambda_per_nm spectral radiance in W/(m^2 sr nm)occupation mean photon number 1/(e^x - 1)x h c/(lambda k T)thermal_de_broglie_wavelengthThe thermal de Broglie wavelength h/sqrt(2 pi m k T) and the quantum concentration 1/lambda^3.
m particle mass [kg]T temperature [K]lambda_th thermal de Broglie wavelength [m]lambda_th_nm the same in nm [nm]n_Q quantum concentration 1/lambda^3 [1/m^3]two_level_thermodynamicsPartition function, Boltzmann populations, mean energy, heat capacity (Schottky) and entropy of a two-level system with gap dE and degeneracies g0, g1.
dE energy gap (eV, J) [J]T temperature [K]g0 degeneracy of the lower level integerg1 degeneracy of the upper level integerZ partition function g0 + g1 e^(-dE/kT)p0 population of the lower levelp1 population of the upper levelratio population ratio N1/N0 (Boltzmann factor)U_eV mean energy above the lower level [eV]C_over_k heat capacity per system in units of kS_over_k entropy per system in units of k15 files, 42.3 MB in total. The largest:
| File | Size |
|---|---|
| model.safetensors | 18.9 MB |
| dist/quantum_physics_ai999x-1.0.0-py3-none-any.whl | 14.5 MB |
| onnx/search/encoder.int8.onnx | 4.9 MB |
| docs/quantum_physics_ai999x_capabilities.pdf | 3 MB |
| onnx/search/index.json | 403.3 KB |
| docs/quantum_physics_ai999x_capabilities.html | 226.6 KB |
| onnx/search/tokenizer.json | 176.4 KB |
| docs/quantum_physics_ai999x_capabilities.md | 167.6 KB |
| README.md | 22.5 KB |
| LICENSE | 11.4 KB |
| evaluation/release_gate.md | 2.3 KB |
| evaluation/KNOWLEDGE_CARD.md | 1.4 KB |
Quantum Physics answers questions in quantum physics: special relativity, photons and matter waves, the hydrogen atom, quantum wells and tunnelling, spin and qubits, quantum gates, nuclear and particle physics and statistical physics. Each answer gives the value with its unit, the steps and the formula in LaTeX.
With pip: pip install quantum-physics-ai999x. The same files are on Hugging Face (hf download AI999X/quantum-physics-ai999x --local-dir quantum-physics-ai999x) and GitHub.
Yes. It is a Python package that runs on an ordinary CPU without a network connection (Python >=3.10); no GPU and no API key are needed.
75 capabilities in 11 topics, including Statistical physics, Special relativity, Quantum wells, Spin and qubits, Hydrogen atom.
Field: Physical Sciences. Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction.
Asked "Lorentz factor: v = 0.8 c", it answers: gamma = 1.66667, beta = 0.8, gamma_minus_1 = 0.666667.
It says so and names what is missing or out of range, instead of guessing a number.