%% CAS_F0028_VERIFY.m — de Broglie relation (p = h/λ) % Pillar : Particle Mechanics % CalRef : Math Appendix §4A, Particle Mechanics Calibration §3C % Chain : constitutive postulate p=h/λ → photon dictionary check → wave number % Hardening: isAlways(..., 'Unknown', 'false') on every symbolic assertion % symvar-based presence checks; solver and has-function avoided clear; clc; fprintf('\n=== CAS F0028 : de Broglie relation ===\n\n'); %% ── Symbols ────────────────────────────────────────────────── syms h_pl lambda_db p_mom real positive % Planck constant, wavelength, momentum syms nu c_light real positive % frequency, speed of light syms E_ph real positive % photon energy syms k_wave real positive % wave number syms hbar real positive % reduced Planck constant %% ── A. Inputs ──────────────────────────────────────────────── % de Broglie postulate: p = h/λ p_deBroglie = h_pl / lambda_db; %% ── Step 1: Constitutive relation structure ───────────────── % p = h/λ → p·λ = h res1 = simplify(p_deBroglie * lambda_db - h_pl); assert(isAlways(res1 == 0, 'Unknown', 'false'), ... 'FAIL Step 1: p·λ ≠ h'); fprintf('Step 1 PASS — p·λ = h (constitutive relation)\n'); %% ── Step 2: Wavelength from momentum ──────────────────────── % λ = h/p lambda_from_p = h_pl / p_mom; % Verify: substituting into p = h/λ recovers p p_check = h_pl / lambda_from_p; res2 = simplify(p_check - p_mom); assert(isAlways(res2 == 0, 'Unknown', 'false'), ... 'FAIL Step 2: h/(h/p) ≠ p'); fprintf('Step 2 PASS — λ = h/p → p = h/λ (round-trip)\n'); %% ── Step 3: Photon dictionary consistency ─────────────────── % Photon: E = hν, λ = c/ν, p = E/c % → p = hν/c = h/(c/ν) = h/λ ✓ E_photon = h_pl * nu; lambda_photon = c_light / nu; p_photon_energy = E_photon / c_light; % p = E/c = hν/c p_photon_deBroglie = h_pl / lambda_photon; % p = h/λ = h/(c/ν) = hν/c res3 = simplify(p_photon_energy - p_photon_deBroglie); assert(isAlways(res3 == 0, 'Unknown', 'false'), ... 'FAIL Step 3: photon E/c ≠ h/λ'); fprintf('Step 3 PASS — Photon: p = E/c = hν/c = h/λ (dictionary consistent)\n'); %% ── Step 4: Wave number form p = ℏk ──────────────────────── % k = 2π/λ, ℏ = h/(2π) % p = h/λ = h·k/(2π) = ℏk p_wavenumber = hbar * k_wave; % Substitute ℏ = h/(2π), k = 2π/λ p_wn_explicit = (h_pl / (2*sym(pi))) * (2*sym(pi) / lambda_db); res4 = simplify(p_wn_explicit - p_deBroglie); assert(isAlways(res4 == 0, 'Unknown', 'false'), ... 'FAIL Step 4: ℏk ≠ h/λ'); fprintf('Step 4 PASS — p = ℏk = h/λ (wave number form)\n'); %% ── Step 5: Energy-momentum for photons ───────────────────── % E = pc (massless) → E = hc/λ = hν % Verify: hc/λ = hν when λ = c/ν E_from_p = p_deBroglie * c_light; % = hc/λ E_from_nu = h_pl * nu; % Substitute λ = c/ν E_sub = subs(E_from_p, lambda_db, c_light/nu); res5 = simplify(E_sub - E_from_nu); assert(isAlways(res5 == 0, 'Unknown', 'false'), ... 'FAIL Step 5: hc/λ ≠ hν at λ=c/ν'); fprintf('Step 5 PASS — E = pc = hc/λ = hν (photon energy-momentum)\n'); %% ── Step 6: Nonrelativistic kinetic energy ────────────────── % T = p²/(2m) = h²/(2mλ²) syms m_part real positive T_kin = p_deBroglie^2 / (2*m_part); T_expected = h_pl^2 / (2*m_part*lambda_db^2); res6 = simplify(T_kin - T_expected); assert(isAlways(res6 == 0, 'Unknown', 'false'), ... 'FAIL Step 6: T ≠ h²/(2mλ²)'); fprintf('Step 6 PASS — T = p²/(2m) = h²/(2mλ²)\n'); %% ── Step 7: Numerical — electron at 100 eV ───────────────── % T = 100 eV → p = √(2mT) % λ = h/p h_val = 6.62607015e-34; m_e = 9.1093837015e-31; eV = 1.602176634e-19; T_100eV = 100 * eV; p_val = sqrt(2 * m_e * T_100eV); lambda_val = h_val / p_val; lambda_expected_nm = 0.1226; % ≈ 0.123 nm for 100 eV electron lambda_calc_nm = lambda_val * 1e9; assert(abs(lambda_calc_nm - lambda_expected_nm) < 0.005, ... 'FAIL Step 7: electron wavelength mismatch'); fprintf('Step 7 PASS — 100 eV electron: λ = %.4f nm (expected ≈ 0.123 nm)\n', lambda_calc_nm); %% ── Step 8: Numerical — photon at visible (550 nm) ───────── % λ = 550 nm, p = h/λ lambda_vis = 550e-9; p_photon_val = h_val / lambda_vis; p_expected = 1.205e-27; % ≈ 1.2×10⁻²⁷ kg·m/s assert(abs(p_photon_val - p_expected)/p_expected < 0.01, ... 'FAIL Step 8: photon momentum mismatch'); fprintf('Step 8 PASS — 550 nm photon: p = %.3e kg·m/s\n', p_photon_val); %% ── Step 9: Cross-block — F0013 wave quantization link ───── % F0013: k_n = 2πn/L (mode quantization) % de Broglie: p = ℏk → p_n = ℏ·2πn/L = hn/L % This is the Bohr–Sommerfeld quantization: p·L = nh syms n_q real positive syms L_box real positive k_n = 2*sym(pi)*n_q / L_box; p_n = (h_pl/(2*sym(pi))) * k_n; res9 = simplify(p_n - h_pl*n_q/L_box); assert(isAlways(res9 == 0, 'Unknown', 'false'), ... 'FAIL Step 9: p_n ≠ hn/L'); % Verify p·L = nh pL = p_n * L_box; res9b = simplify(pL - n_q*h_pl); assert(isAlways(res9b == 0, 'Unknown', 'false'), ... 'FAIL Step 9b: p·L ≠ nh'); fprintf('Step 9 PASS — p_n = hn/L, p·L = nh (consistent with F0013)\n'); %% ── Step 10: Self-test — wrong relation p = h·λ ──────────── % Wrong: p = h·λ (multiply instead of divide) p_wrong = h_pl * lambda_db; res_wrong = simplify(p_wrong - p_deBroglie); assert(~isAlways(res_wrong == 0, 'Unknown', 'false'), ... 'FAIL Step 10a: wrong relation not detected'); fprintf('Step 10a PASS — Wrong p=h·λ detected as incorrect\n'); % Quantify residual: p_wrong − p_correct = hλ − h/λ = h(λ − 1/λ) = h(λ²−1)/λ res_quant = simplify(res_wrong - h_pl*(lambda_db - 1/lambda_db)); assert(isAlways(res_quant == 0, 'Unknown', 'false'), ... 'FAIL Step 10b: wrong residual not quantified'); fprintf('Step 10b PASS — Wrong residual = h(λ−1/λ) (quantified)\n'); %% ── Summary ───────────────────────────────────────────────── fprintf('\n✓ ALL 11 ASSERTIONS PASSED — F0028 audit complete.\n'); fprintf(' Chain: p=h/λ (postulate) → photon dictionary → ℏk form → quantization\n'); fprintf(' Cross-ref: F0013 (mode quantization), F0019 (photon E=hν)\n');