+ i^RIHt^RIHt^RIHt^RIHt^RIH t ^RI H t ^RI H t RRltRR ltRR ltR ^]!R 43R ltR#))Float)S) factorial)exp)sqrt)assoc_laguerre)Ynmc\\WW#.4wrr#W, ^, p^V, p^V,W,, p\\^4W,, ^,\V4,^V,\W,4,, 4pWvV,,\ V^V,^,V4P 4,\ V)^, 4,#)a Returns the Hydrogen radial wavefunction R_{nl}. Parameters ========== n : integer Principal Quantum Number which is an integer with possible values as 1, 2, 3, 4,... l : integer ``l`` is the Angular Momentum Quantum Number with values ranging from 0 to ``n-1``. 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.abc import 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 )maprrrrexpandr)nlrZn_rar0Cs&&&& v/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/physics/hydrogen.pyR_nlr sNQq %JA! %!)C !A Q!%B adACj1_y~ -1Yqu5E1E FGA 1u9~c1Q37B7>>@ @3s1u: MMc v\\WW#WEV.4wrr#rEpVP'dV^8d \R4hVP'dW8g \R4hVP'd\ V4V8:g \R4h\ WW64\ WWT4PRR7,#)a Returns the Hydrogen wave function psi_{nlm}. It's the product of the radial wavefunction R_{nl} and the spherical harmonic Y_{l}^{m}. Parameters ========== n : integer Principal Quantum Number which is an integer with possible values as 1, 2, 3, 4,... l : integer ``l`` is the Angular Momentum Quantum Number with values ranging from 0 to ``n-1``. m : integer ``m`` is the Magnetic Quantum Number with values ranging from ``-l`` to ``l``. 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", 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 'n' must be positive integer'n' must be greater than 'l'z|'m'| must be less or equal 'l'T)func)r r is_integer ValueErrorabsrr r )r rmrphithetars&&&&&&&rPsi_nlmr"_sp!$AaC'B CA!A|||A788|||QU788|||SVq[:;; a Ce188d8C CCrc\V4\V4rVP'dV^8d \R4hV^,)^V^,,, #)a Returns the energy of the state (n, l) in Hartree atomic units. The energy does not depend on "l". Parameters ========== n : integer Principal Quantum Number which is an integer with possible values as 1, 2, 3, 4,... Z : Atomic number (1 for Hydrogen, 2 for Helium, ...) Examples ======== >>> from sympy.physics.hydrogen import E_nl >>> from sympy.abc import n, Z >>> E_nl(n, Z) -Z**2/(2*n**2) >>> E_nl(1) -1/2 >>> E_nl(2) -1/8 >>> E_nl(3) -1/18 >>> E_nl(3, 47) -2209/18 r)rrr)r rs&&rE_nlr$sG@ Q41q|||Q788 qD5!AqD&>rTz 137.035999037c\\WW4.4wrr4V^8g \R4hW8g \R4hV^8XdVRJd \R4hV'd V)^, pMV)p\V^,V^,V^,, , 4pV^,\^V^,W,V,^,, V^,, ,4, V^,, #)a[ 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. Parameters ========== n : integer Principal Quantum Number which is an integer with possible values as 1, 2, 3, 4,... l : integer ``l`` is the Angular Momentum Quantum Number with values ranging from 0 to ``n-1``. 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 https://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 z'l' must be positive or zerorFzSpin must be up for l==0.)r rrr)r rspin_uprcskappabetas&&&&& r E_nl_diracr*sfQq %JA! F788 E788 Q7e#455a  AqDAI% &D a4QAqzD0144QT99: :QT AArN))sympy.core.numbersrsympy.core.singletonr(sympy.functions.combinatorial.factorialsr&sympy.functions.elementary.exponentialr(sympy.functions.elementary.miscellaneousr#sympy.functions.special.polynomialsr+sympy.functions.special.spherical_harmonicsr rr"r$r*rrr4sE$">69>;RNjADH#L"Q%*@@Br