"Fine structure 1" "Verify equation (1)" x = alpha / (A + sqrt(B^2 - alpha^2)) f = 1 / sqrt(1 + x^2) f = taylor(f,alpha,4) f = eval(f, A, n - j - 1/2, B, j + 1/2) y = 1 - alpha^2 / (2 n^2) - alpha^4 / (2 n^3 (j + 1/2)) + 3 alpha^4 / (8 n^4) check(f == y) "ok" "Verify equation (2)" E = -mu c^2 alpha^2 / (2 n^2) (1 + alpha^2 / (n (j + 1/2)) - 3 alpha^2 / (4 n^2)) check(-mu c^2 (1 - f) == E) "ok" -- CODATA Internationally recommended 2022 values -- https://physics.nist.gov/cuu/Constants/ alpha = 7.2973525643 10^(-3) c = 299792458.0 meter / second e = 1.602176634 10^(-19) coulomb me = 9.1093837139 10^(-31) kilogram mp = 1.67262192595 10^(-27) kilogram joule = kilogram meter^2 / second^2 eV = 1/e coulomb / joule "eV" -- eV per joule mu = me mp / (me + mp) -- reduced electron mass -mu c^2 alpha^2 / 2 eV Run