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from sympy.core.numbers import Float
from sympy.core.singleton import S
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.special.polynomials import assoc_laguerre
from sympy.functions.special.spherical_harmonics import Ynm
def R_nl(n, l, r, Z=1):
"""
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
"""
# sympify arguments
n, l, r, Z = map(S, [n, l, r, 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 Psi_nlm(n, l, m, r, phi, theta, Z=1):
"""
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
"""
# sympify arguments
n, l, m, r, phi, theta, Z = map(S, [n, l, m, r, phi, theta, Z])
# check if values for n,l,m make physically sense
if n.is_integer and n < 1:
raise ValueError("'n' must be positive integer")
if l.is_integer and not (n > l):
raise ValueError("'n' must be greater than 'l'")
if m.is_integer and not (abs(m) <= l):
raise ValueError("|'m'| must be less or equal 'l'")
# return the hydrogen wave function
return R_nl(n, l, r, Z)*Ynm(l, m, theta, phi).expand(func=True)
def E_nl(n, Z=1):
"""
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
"""
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.
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
"""
n, l, Z, c = map(S, [n, l, Z, c])
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
beta = sqrt(skappa**2 - Z**2/c**2)
return c**2/sqrt(1 + Z**2/(n + skappa + beta)**2/c**2) - c**2
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