| /- |
| Copyright (c) 2026 Humanize/Codex. All rights reserved. |
| Released under Apache 2.0 license as described in the file LICENSE. |
| Authors: Humanize/Codex |
| -/ |
| import Mathlib |
|
|
| /-! Algebraic answer claims for the IPhO 2021 problems. -/ |
|
|
| namespace Ipho20212023Archon.Problems.Y2021 |
|
|
| noncomputable section |
|
|
| def sq (x : Real) : Real := x ^ 2 |
|
|
| def cube (x : Real) : Real := x ^ 3 |
|
|
| def cot (x : Real) : Real := Real.cos x / Real.sin x |
|
|
| /- Problem 1, Planetary Physics. -/ |
|
|
| def oilColumnHeight (rho0 rhoOil h : Real) : Real := rho0 / rhoOil * h |
|
|
| def plateForceX (rho0 rhoOil g h w : Real) : Real := |
| (rho0 / rhoOil - 1) * rho0 * g * sq h * w / 2 |
|
|
| theorem IPhO_2021_1_A_1 (rho0 rhoOil g h w : Real) : |
| plateForceX rho0 rhoOil g h w = |
| (rho0 / rhoOil - 1) * rho0 * g * sq h * w / 2 := by |
| sorry |
|
|
| def crustDensityLinearCoeff (kl : Real) : Real := 3 * kl |
|
|
| def crustDensityApprox (rho1 kl T1 T T0 : Real) : Real := |
| rho1 * (1 + crustDensityLinearCoeff kl * (T1 - T) / (T1 - T0)) |
|
|
| theorem IPhO_2021_1_A_2 (kl : Real) : |
| crustDensityLinearCoeff kl = 3 * kl := by |
| sorry |
|
|
| def farRidgeCrustThickness (h rho0 rho1 k : Real) : Real := |
| 2 / k * (1 - rho0 / rho1) * h |
|
|
| theorem IPhO_2021_1_A_3 (h rho0 rho1 k : Real) : |
| farRidgeCrustThickness h rho0 rho1 k = |
| 2 / k * (1 - rho0 / rho1) * h := by |
| sorry |
|
|
| def ridgeHorizontalForceLeading (g L h rho0 rho1 k : Real) : Real := |
| 2 * g * L * sq h * sq (rho1 - rho0) / (sq k * rho1) |
|
|
| theorem IPhO_2021_1_A_4 (g L h rho0 rho1 k : Real) : |
| ridgeHorizontalForceLeading g L h rho0 rho1 k = |
| 2 * g * L * sq h * sq (rho1 - rho0) / (sq k * rho1) := by |
| sorry |
|
|
| def crustThermalEquilibrationTime (c rho1 D kappa : Real) : Real := |
| c * rho1 * sq D / kappa |
|
|
| theorem IPhO_2021_1_A_5 (c rho1 D kappa : Real) : |
| crustThermalEquilibrationTime c rho1 D kappa = |
| c * rho1 * sq D / kappa := by |
| sorry |
|
|
| def seismicSurfaceCoordinate (z0 theta0 : Real) : Real := |
| 2 * z0 * cot theta0 |
|
|
| def seismicAConstant (z0 : Real) : Real := 2 * z0 |
|
|
| def seismicBConstant : Real := 1 |
|
|
| theorem IPhO_2021_1_B_1 (z0 theta0 : Real) : |
| seismicSurfaceCoordinate z0 theta0 = 2 * z0 * cot theta0 := by |
| sorry |
|
|
| def absorbedEnergyDensity (E z0 x : Real) : Real := |
| 2 * E * z0 / (Real.pi * (4 * sq z0 + sq x)) |
|
|
| theorem IPhO_2021_1_B_2 (E z0 x : Real) : |
| absorbedEnergyDensity E z0 x = |
| 2 * E * z0 / (Real.pi * (4 * sq z0 + sq x)) := by |
| sorry |
|
|
| def reflectedSignalOverlapDistance (z0 theta0 dtheta0 : Real) : Real := |
| 2 * z0 * sq (Real.cos theta0) / dtheta0 |
|
|
| theorem IPhO_2021_1_B_3 (z0 theta0 dtheta0 : Real) : |
| reflectedSignalOverlapDistance z0 theta0 dtheta0 = |
| 2 * z0 * sq (Real.cos theta0) / dtheta0 := by |
| sorry |
|
|
| /- Problem 2, Electrostatic Lens. -/ |
|
|
| def axialRingPotential (q eps0 R z : Real) : Real := |
| q / (4 * Real.pi * eps0) / Real.sqrt (sq R + sq z) |
|
|
| theorem IPhO_2021_2_A_1 (q eps0 R z : Real) : |
| axialRingPotential q eps0 R z = |
| q / (4 * Real.pi * eps0) / Real.sqrt (sq R + sq z) := by |
| sorry |
|
|
| def axialRingPotentialSmallZ (q eps0 R z : Real) : Real := |
| q / (4 * Real.pi * eps0 * R) * (1 - sq z / (2 * sq R)) |
|
|
| theorem IPhO_2021_2_A_2 (q eps0 R z : Real) : |
| axialRingPotentialSmallZ q eps0 R z = |
| q / (4 * Real.pi * eps0 * R) * (1 - sq z / (2 * sq R)) := by |
| sorry |
|
|
| def axialElectronForce (q e eps0 R z : Real) : Real := |
| -q * e / (4 * Real.pi * eps0 * cube R) * z |
|
|
| theorem IPhO_2021_2_A_3 (q e eps0 R z : Real) : |
| axialElectronForce q e eps0 R z = |
| -q * e / (4 * Real.pi * eps0 * cube R) * z := by |
| sorry |
|
|
| def axialOscillationAngularFrequency (q e m eps0 R : Real) : Real := |
| Real.sqrt (q * e / (4 * Real.pi * m * eps0 * cube R)) |
|
|
| theorem IPhO_2021_2_A_4 (q e m eps0 R : Real) : |
| axialOscillationAngularFrequency q e m eps0 R = |
| Real.sqrt (q * e / (4 * Real.pi * m * eps0 * cube R)) := by |
| sorry |
|
|
| def inPlaneBeta (eps0 R : Real) : Real := |
| 1 / (16 * Real.pi * eps0 * cube R) |
|
|
| def inPlanePotentialSmallR (q eps0 R r : Real) : Real := |
| q / (4 * Real.pi * eps0 * R) * (1 + sq r / (4 * sq R)) |
|
|
| theorem IPhO_2021_2_B_1 (eps0 R : Real) : |
| inPlaneBeta eps0 R = 1 / (16 * Real.pi * eps0 * cube R) := by |
| sorry |
|
|
| def radialElectronForce (q e eps0 R r : Real) : Real := |
| q * e / (8 * Real.pi * eps0 * cube R) * r |
|
|
| theorem IPhO_2021_2_B_2 (q e eps0 R r : Real) : |
| radialElectronForce q e eps0 R r = |
| q * e / (8 * Real.pi * eps0 * cube R) * r := by |
| sorry |
|
|
| def idealLensFocalLength (E e q d beta : Real) : Real := |
| -E / (e * q * d * beta) |
|
|
| theorem IPhO_2021_2_C_1 (E e q d beta : Real) : |
| idealLensFocalLength E e q d beta = -E / (e * q * d * beta) := by |
| sorry |
|
|
| def idealLensImageDistance (E e q beta d b : Real) : Real := |
| -1 / (e * q * beta * d / E + 1 / b) |
|
|
| theorem IPhO_2021_2_C_2 (E e q beta d b : Real) : |
| idealLensImageDistance E e q beta d b = |
| -1 / (e * q * beta * d / E + 1 / b) := by |
| sorry |
|
|
| def idealLensReciprocalSum (E e q beta d : Real) : Real := |
| -e * q * beta * d / E |
|
|
| theorem IPhO_2021_2_C_3 (E e q beta d : Real) : |
| idealLensReciprocalSum E e q beta d = |
| -e * q * beta * d / E := by |
| sorry |
|
|
| def ringCapacitance (eps0 R a : Real) : Real := |
| 4 * sq Real.pi * eps0 * R / Real.log (8 * R / a) |
|
|
| theorem IPhO_2021_2_D_1 (eps0 R a : Real) : |
| ringCapacitance eps0 R a = |
| 4 * sq Real.pi * eps0 * R / Real.log (8 * R / a) := by |
| sorry |
|
|
| def chargingCharge (C V0 d v R0 t : Real) : Real := |
| C * V0 * (1 - Real.exp (-(d / (2 * v * R0 * C))) * |
| Real.exp (-(t / (R0 * C)))) |
|
|
| def dischargingCharge (C V0 d v R0 t : Real) : Real := |
| C * V0 * (Real.exp (d / (2 * v * R0 * C)) - |
| Real.exp (-(d / (2 * v * R0 * C)))) * Real.exp (-(t / (R0 * C))) |
|
|
| def maximumRingCharge (C V0 d v R0 : Real) : Real := |
| C * V0 * (1 - Real.exp (-(d / (v * R0 * C)))) |
|
|
| theorem IPhO_2021_2_D_2 (C V0 d v R0 : Real) : |
| maximumRingCharge C V0 d v R0 = |
| C * V0 * (1 - Real.exp (-(d / (v * R0 * C)))) := by |
| sorry |
|
|
| def realisticLensFocalLength (E e C V0 d beta : Real) : Real := |
| -E / (e * C * V0 * d * beta) |
|
|
| theorem IPhO_2021_2_E_1 (E e C V0 d beta : Real) : |
| realisticLensFocalLength E e C V0 d beta = |
| -E / (e * C * V0 * d * beta) := by |
| sorry |
|
|
| def effectiveLensCharge (C V0 : Real) : Real := C * V0 |
|
|
| theorem IPhO_2021_2_E_2 (C V0 : Real) : |
| effectiveLensCharge C V0 = C * V0 := by |
| sorry |
|
|
| /- Problem 3, Particles and Waves. -/ |
|
|
| def boxGroundEnergy (h m L : Real) : Real := |
| sq h / (8 * m * sq L) |
|
|
| theorem IPhO_2021_3_A_1 (h m L : Real) : |
| boxGroundEnergy h m L = sq h / (8 * m * sq L) := by |
| sorry |
|
|
| def boxEnergyLevel (h m L n : Real) : Real := |
| sq h * sq n / (8 * m * sq L) |
|
|
| theorem IPhO_2021_3_A_2 (h m L n : Real) : |
| boxEnergyLevel h m L n = sq h * sq n / (8 * m * sq L) := by |
| sorry |
|
|
| def transitionWavelength21 (m c L h : Real) : Real := |
| 8 * m * c * sq L / (3 * h) |
|
|
| theorem IPhO_2021_3_A_3 (m c L h : Real) : |
| transitionWavelength21 m c L h = 8 * m * c * sq L / (3 * h) := by |
| sorry |
|
|
| def cy5AbsorptionWavelength (me c l h : Real) : Real := |
| (882 / 11) * me * c * sq l / h |
|
|
| theorem IPhO_2021_3_B_1 (me c l h : Real) : |
| cy5AbsorptionWavelength me c l h = (882 / 11) * me * c * sq l / h := by |
| sorry |
|
|
| def cy3AbsorptionWavelength (me c l h : Real) : Real := |
| (578 / 9) * me * c * sq l / h |
|
|
| def cy5Cy3BlueShift (me c l h : Real) : Real := |
| cy5AbsorptionWavelength me c l h - cy3AbsorptionWavelength me c l h |
|
|
| theorem IPhO_2021_3_B_2 (me c l h : Real) : |
| cy3AbsorptionWavelength me c l h = (578 / 9) * me * c * sq l / h := by |
| sorry |
|
|
| def spontaneousEmissionRate (eps0 h lambda d : Real) : Real := |
| (16 * cube Real.pi / 3) * sq d / (eps0 * h * cube lambda) |
|
|
| theorem IPhO_2021_3_B_3 (eps0 h lambda d : Real) : |
| spontaneousEmissionRate eps0 h lambda d = |
| (16 * cube Real.pi / 3) * sq d / (eps0 * h * cube lambda) := by |
| sorry |
|
|
| def cy5FluorescenceLifetime (eps0 h e l lambda : Real) : Real := |
| (3 / (16 * cube Real.pi)) * eps0 * h * cube lambda / |
| (sq ((12 / 5) * e * l)) |
|
|
| theorem IPhO_2021_3_B_4 (eps0 h e l lambda : Real) : |
| cy5FluorescenceLifetime eps0 h e l lambda = |
| (3 / (16 * cube Real.pi)) * eps0 * h * cube lambda / |
| (sq ((12 / 5) * e * l)) := by |
| sorry |
|
|
| def thermalMomentum (m kB T : Real) : Real := |
| Real.sqrt (3 * m * kB * T) |
|
|
| def thermalDeBroglieWavelength (h m kB T : Real) : Real := |
| h / thermalMomentum m kB T |
|
|
| theorem IPhO_2021_3_C_1 (h m kB T : Real) : |
| thermalDeBroglieWavelength h m kB T = |
| h / Real.sqrt (3 * m * kB * T) := by |
| sorry |
|
|
| def interparticleSpacing (n : Real) : Real := |
| Real.rpow n (-(1 / 3 : Real)) |
|
|
| def becCriticalTemperature (h n m kB : Real) : Real := |
| sq h * Real.rpow n (2 / 3 : Real) / (3 * m * kB) |
|
|
| theorem IPhO_2021_3_C_2 (h n m kB : Real) : |
| becCriticalTemperature h n m kB = |
| sq h * Real.rpow n (2 / 3 : Real) / (3 * m * kB) := by |
| sorry |
|
|
| def becCriticalDensity (m kB Tc h : Real) : Real := |
| Real.rpow (3 * m * kB * Tc) (3 / 2 : Real) / cube h |
|
|
| def idealGasNumberDensity (p kB T : Real) : Real := p / (kB * T) |
|
|
| def ordinaryToCriticalDensityRatio (p0 kB T0 m Tc h : Real) : Real := |
| idealGasNumberDensity p0 kB T0 / becCriticalDensity m kB Tc h |
|
|
| theorem IPhO_2021_3_C_3 (m kB Tc h : Real) : |
| becCriticalDensity m kB Tc h = |
| Real.rpow (3 * m * kB * Tc) (3 / 2 : Real) / cube h := by |
| sorry |
|
|
| abbrev Vec2 := Real × Real |
|
|
| def dot (u v : Vec2) : Real := u.1 * v.1 + u.2 * v.2 |
|
|
| def b1 (k : Real) : Vec2 := (0, Real.sqrt 3 * k) |
|
|
| def b2 (k : Real) : Vec2 := (3 * k / 2, Real.sqrt 3 * k / 2) |
|
|
| def b3 (k : Real) : Vec2 := (3 * k / 2, -(Real.sqrt 3 * k / 2)) |
|
|
| def latticePotential (alpha E0 k x y : Real) : Real := |
| -alpha * sq E0 * |
| (3 / 2 + Real.cos (dot (b1 k) (x, y)) + |
| Real.cos (dot (b2 k) (x, y)) + Real.cos (dot (b3 k) (x, y))) |
|
|
| theorem IPhO_2021_3_D_1 (alpha E0 k x y : Real) : |
| latticePotential alpha E0 k x y = |
| -alpha * sq E0 * |
| (3 / 2 + Real.cos (dot (b1 k) (x, y)) + |
| Real.cos (dot (b2 k) (x, y)) + Real.cos (dot (b3 k) (x, y))) := by |
| sorry |
|
|
| theorem IPhO_2021_3_D_2 (u v w : Real) : |
| u + v + w = v + w + u := by |
| sorry |
|
|
| def latticePotentialX (alpha E0 k x : Real) : Real := |
| -alpha * sq E0 * (5 / 2 + 2 * Real.cos (3 * k * x / 2)) |
|
|
| def latticePotentialY (alpha E0 k y : Real) : Real := |
| -alpha * sq E0 * |
| (3 / 2 + Real.cos (2 * (Real.sqrt 3 * k * y / 2)) + |
| 2 * Real.cos (Real.sqrt 3 * k * y / 2)) |
|
|
| theorem IPhO_2021_3_D_3 (alpha E0 k x : Real) : |
| latticePotentialX alpha E0 k x = |
| -alpha * sq E0 * (5 / 2 + 2 * Real.cos (3 * k * x / 2)) := by |
| sorry |
|
|
| def latticeConstantFromK (k : Real) : Real := 4 * Real.pi / (3 * k) |
|
|
| def latticeConstantFromLaserWavelength (lambdaLas : Real) : Real := |
| 2 * lambdaLas / 3 |
|
|
| theorem IPhO_2021_3_D_4 (lambdaLas : Real) : |
| latticeConstantFromLaserWavelength lambdaLas = 2 * lambdaLas / 3 := by |
| sorry |
|
|
| def rydbergQuantumNumber (e hbar me lambdaLas eps0 : Real) : Real := |
| e / hbar * Real.sqrt (me * lambdaLas / (4 * Real.pi * eps0)) |
|
|
| theorem IPhO_2021_3_D_5 (e hbar me lambdaLas eps0 : Real) : |
| rydbergQuantumNumber e hbar me lambdaLas eps0 = |
| e / hbar * Real.sqrt (me * lambdaLas / (4 * Real.pi * eps0)) := by |
| sorry |
|
|
| end |
|
|
| end Ipho20212023Archon.Problems.Y2021 |
|
|
