def test_laguerre():
alpha = Symbol("alpha")
# generalized Laguerre polynomials:
assert laguerre_l(0, alpha, x) == 1
assert laguerre_l(1, alpha, x) == -x + alpha + 1
assert laguerre_l(2, alpha,
x).expand() == (x**2 / 2 - (alpha + 2) * x +
(alpha + 2) * (alpha + 1) / 2).expand()
assert laguerre_l(
3, alpha,
x).expand() == (-x**3 / 6 + (alpha + 3) * x**2 / 2 - (alpha + 2) *
(alpha + 3) * x / 2 + (alpha + 1) * (alpha + 2) *
(alpha + 3) / 6).expand()
# Laguerre polynomials:
assert laguerre_l(0, 0, x) == 1
assert laguerre_l(1, 0, x) == 1 - x
assert laguerre_l(2, 0, x).expand() == 1 - 2 * x + x**2 / 2
assert laguerre_l(3, 0, x).expand() == 1 - 3 * x + 3 * x**2 / 2 - x**3 / 6
# Test the lowest 10 polynomials with laguerre_poly, to make sure that it
# works:
for i in range(10):
assert laguerre_l(i, 0, x).expand() == laguerre_poly(i, x)
def test_laguerre():
alpha = Symbol("alpha")
# generalized Laguerre polynomials:
assert laguerre_l(0, alpha, x) == 1
assert laguerre_l(1, alpha, x) == -x + alpha + 1
assert laguerre_l(2, alpha, x).expand() == (x**2/2 - (alpha+2)*x + (alpha+2)*(alpha+1)/2).expand()
assert laguerre_l(3, alpha, x).expand() == (-x**3/6 + (alpha+3)*x**2/2 - (alpha+2)*(alpha+3)*x/2 + (alpha+1)*(alpha+2)*(alpha+3)/6).expand()
# Laguerre polynomials:
assert laguerre_l(0, 0, x) == 1
assert laguerre_l(1, 0, x) == 1 - x
assert laguerre_l(2, 0, x).expand() == 1 - 2*x + x**2/2
assert laguerre_l(3, 0, x).expand() == 1 - 3*x + 3*x**2/2 - x**3/6
# Test the lowest 10 polynomials with laguerre_poly, to make sure that it
# works:
for i in range(10):
assert laguerre_l(i, 0, x).expand() == laguerre_poly(i, x)
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*(Z**3)**(1/2)*exp(-Z*r)
>>> R_nl(2, 0, r, Z)
2**(1/2)*(Z**3)**(1/2)*(2 - Z*r)*exp(-Z*r/2)/4
>>> R_nl(2, 1, r, Z)
Z*r*6**(1/2)*(Z**3)**(1/2)*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)
2**(1/2)*(2 - r)*exp(-r/2)/4
>>> R_nl(3, 0, r)
2*3**(1/2)*(3 - 2*r + 2*r**2/9)*exp(-r/3)/27
For Silver atom, you would use Z=47::
>>> R_nl(1, 0, r, Z=47)
94*47**(1/2)*exp(-47*r)
>>> R_nl(2, 0, r, Z=47)
47*94**(1/2)*(2 - 47*r)*exp(-47*r/2)/4
>>> R_nl(3, 0, r, Z=47)
94*141**(1/2)*(3 - 94*r + 4418*r**2/9)*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 * laguerre_l(n_r, 2 * l + 1, r0).expand() * exp(-r0 / 2)
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 * laguerre_l(n_r, 2 * l + 1, r0).expand() * exp(-r0 / 2)
