# sympy/sympy

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 from __future__ import print_function, division from sympy import factorial, sqrt, exp, S, assoc_laguerre, Float def R_nl(n, l, r, Z=1): """ Returns the Hydrogen radial wavefunction R_{nl}. n, l quantum numbers 'n' and 'l' r radial coordinate Z atomic number (1 for Hydrogen, 2 for Helium, ...) Everything is in Hartree atomic units. Examples ======== >>> from sympy.physics.hydrogen import R_nl >>> from sympy import var >>> var("r Z") (r, Z) >>> R_nl(1, 0, r, Z) 2*sqrt(Z**3)*exp(-Z*r) >>> R_nl(2, 0, r, Z) sqrt(2)*(-Z*r + 2)*sqrt(Z**3)*exp(-Z*r/2)/4 >>> R_nl(2, 1, r, Z) sqrt(6)*Z*r*sqrt(Z**3)*exp(-Z*r/2)/12 For Hydrogen atom, you can just use the default value of Z=1: >>> R_nl(1, 0, r) 2*exp(-r) >>> R_nl(2, 0, r) sqrt(2)*(-r + 2)*exp(-r/2)/4 >>> R_nl(3, 0, r) 2*sqrt(3)*(2*r**2/9 - 2*r + 3)*exp(-r/3)/27 For Silver atom, you would use Z=47: >>> R_nl(1, 0, r, Z=47) 94*sqrt(47)*exp(-47*r) >>> R_nl(2, 0, r, Z=47) 47*sqrt(94)*(-47*r + 2)*exp(-47*r/2)/4 >>> R_nl(3, 0, r, Z=47) 94*sqrt(141)*(4418*r**2/9 - 94*r + 3)*exp(-47*r/3)/27 The normalization of the radial wavefunction is: >>> from sympy import integrate, oo >>> integrate(R_nl(1, 0, r)**2 * r**2, (r, 0, oo)) 1 >>> integrate(R_nl(2, 0, r)**2 * r**2, (r, 0, oo)) 1 >>> integrate(R_nl(2, 1, r)**2 * r**2, (r, 0, oo)) 1 It holds for any atomic number: >>> integrate(R_nl(1, 0, r, Z=2)**2 * r**2, (r, 0, oo)) 1 >>> integrate(R_nl(2, 0, r, Z=3)**2 * r**2, (r, 0, oo)) 1 >>> integrate(R_nl(2, 1, r, Z=4)**2 * r**2, (r, 0, oo)) 1 """ # sympify arguments n, l, r, Z = S(n), S(l), S(r), S(Z) # radial quantum number n_r = n - l - 1 # rescaled "r" a = 1/Z # Bohr radius r0 = 2 * r / (n * a) # normalization coefficient C = sqrt((S(2)/(n*a))**3 * factorial(n_r) / (2*n*factorial(n + l))) # This is an equivalent normalization coefficient, that can be found in # some books. Both coefficients seem to be the same fast: # C = S(2)/n**2 * sqrt(1/a**3 * factorial(n_r) / (factorial(n+l))) return C * r0**l * assoc_laguerre(n_r, 2*l + 1, r0).expand() * exp(-r0/2) def E_nl(n, Z=1): """ Returns the energy of the state (n, l) in Hartree atomic units. The energy doesn't depend on "l". Examples ======== >>> from sympy import var >>> from sympy.physics.hydrogen import E_nl >>> var("n Z") (n, Z) >>> E_nl(n, Z) -Z**2/(2*n**2) >>> E_nl(1) -1/2 >>> E_nl(2) -1/8 >>> E_nl(3) -1/18 >>> E_nl(3, 47) -2209/18 """ n, Z = S(n), S(Z) if n.is_integer and (n < 1): raise ValueError("'n' must be positive integer") return -Z**2/(2*n**2) def E_nl_dirac(n, l, spin_up=True, Z=1, c=Float("137.035999037")): """ Returns the relativistic energy of the state (n, l, spin) in Hartree atomic units. The energy is calculated from the Dirac equation. The rest mass energy is *not* included. n, l quantum numbers 'n' and 'l' spin_up True if the electron spin is up (default), otherwise down Z atomic number (1 for Hydrogen, 2 for Helium, ...) c speed of light in atomic units. Default value is 137.035999037, taken from: http://arxiv.org/abs/1012.3627 Examples ======== >>> from sympy.physics.hydrogen import E_nl_dirac >>> E_nl_dirac(1, 0) -0.500006656595360 >>> E_nl_dirac(2, 0) -0.125002080189006 >>> E_nl_dirac(2, 1) -0.125000416028342 >>> E_nl_dirac(2, 1, False) -0.125002080189006 >>> E_nl_dirac(3, 0) -0.0555562951740285 >>> E_nl_dirac(3, 1) -0.0555558020932949 >>> E_nl_dirac(3, 1, False) -0.0555562951740285 >>> E_nl_dirac(3, 2) -0.0555556377366884 >>> E_nl_dirac(3, 2, False) -0.0555558020932949 """ if not (l >= 0): raise ValueError("'l' must be positive or zero") if not (n > l): raise ValueError("'n' must be greater than 'l'") if (l == 0 and spin_up is False): raise ValueError("Spin must be up for l==0.") # skappa is sign*kappa, where sign contains the correct sign if spin_up: skappa = -l - 1 else: skappa = -l c = S(c) beta = sqrt(skappa**2 - Z**2/c**2) return c**2/sqrt(1 + Z**2/(n + skappa + beta)**2/c**2) - c**2