ó ¡¼™\c@ s”ddlmZmZddlmZmZmZmZmZm Z ddl m Z dd„Z dd„Z dd„Zede dƒd „Zd S( iÿÿÿÿ(tprint_functiontdivision(t factorialtsqrttexptStassoc_laguerretFloat(tYnmicC s×t|ƒt|ƒt|ƒt|ƒf\}}}}||d}d|}d|||}ttdƒ||dt|ƒd|t||ƒƒ}|||t|d|d|ƒjƒt| dƒS(s! 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)*(2 - r)*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)*(2 - 47*r)*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 iii(RRRRtexpandR(tntltrtZtn_rtatr0tC((s5lib/python2.7/site-packages/sympy/physics/hydrogen.pytR_nls B6 >cC st|ƒt|ƒt|ƒt|ƒt|ƒt|ƒt|ƒf\}}}}}}}|jr~|dkr~tdƒ‚n|jr£||k r£tdƒ‚n|jrÎt|ƒ|k rÎtdƒ‚nt||||ƒt||||ƒjdtƒS(s Returns the Hydrogen wave function psi_{nlm}. It's the product of the radial wavefunction R_{nl} and the spherical harmonic Y_{l}^{m}. n, l, m quantum numbers 'n', 'l' and 'm' r radial coordinate phi azimuthal angle theta polar angle Z atomic number (1 for Hydrogen, 2 for Helium, ...) Everything is in Hartree atomic units. Examples ======== >>> from sympy.physics.hydrogen import Psi_nlm >>> from sympy import Symbol >>> r=Symbol("r", real=True, positive=True) >>> phi=Symbol("phi", real=True) >>> theta=Symbol("theta", real=True) >>> Z=Symbol("Z", positive=True, integer=True, nonzero=True) >>> Psi_nlm(1,0,0,r,phi,theta,Z) Z**(3/2)*exp(-Z*r)/sqrt(pi) >>> Psi_nlm(2,1,1,r,phi,theta,Z) -Z**(5/2)*r*exp(I*phi)*exp(-Z*r/2)*sin(theta)/(8*sqrt(pi)) Integrating the absolute square of a hydrogen wavefunction psi_{nlm} over the whole space leads 1. The normalization of the hydrogen wavefunctions Psi_nlm is: >>> from sympy import integrate, conjugate, pi, oo, sin >>> wf=Psi_nlm(2,1,1,r,phi,theta,Z) >>> abs_sqrd=wf*conjugate(wf) >>> jacobi=r**2*sin(theta) >>> integrate(abs_sqrd*jacobi, (r,0,oo), (phi,0,2*pi), (theta,0,pi)) 1 is'n' must be positive integers'n' must be greater than 'l's|'m'| must be less or equal 'l'tfunc(Rt is_integert ValueErrortabsRRR tTrue(R R tmR tphitthetaR ((s5lib/python2.7/site-packages/sympy/physics/hydrogen.pytPsi_nlmVs.ZcC sRt|ƒt|ƒ}}|jr=|dkr=tdƒ‚n|d d|dS(sŒ 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 is'n' must be positive integeri(RRR(R R ((s5lib/python2.7/site-packages/sympy/physics/hydrogen.pytE_nl‘ss 137.035999037cC sÞ|dkstdƒ‚n||ks6tdƒ‚n|dkr]|tkr]tdƒ‚n|rq| d}n| }t|ƒ}t|d|d|dƒ}|dtd|d|||d|dƒ|dS(s 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 is'l' must be positive or zeros'n' must be greater than 'l'sSpin must be up for l==0.ii(RtFalseRR(R R tspin_upR tctskappatbeta((s5lib/python2.7/site-packages/sympy/physics/hydrogen.pyt E_nl_dirac°s,    N(t __future__RRtsympyRRRRRRt+sympy.functions.special.spherical_harmonicsRRRRRR"(((s5lib/python2.7/site-packages/sympy/physics/hydrogen.pyts . O ;