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from sympy.core.numbers import (I, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.integrals.integrals import integrate
from sympy.simplify.simplify import simplify
from sympy.physics.hydrogen import R_nl, E_nl, E_nl_dirac, Psi_nlm
from sympy.testing.pytest import raises
n, r, Z = symbols('n r Z')
def feq(a, b, max_relative_error=1e-12, max_absolute_error=1e-12):
a = float(a)
b = float(b)
# if the numbers are close enough (absolutely), then they are equal
if abs(a - b) < max_absolute_error:
return True
# if not, they can still be equal if their relative error is small
if abs(b) > abs(a):
relative_error = abs((a - b)/b)
else:
relative_error = abs((a - b)/a)
return relative_error <= max_relative_error
def test_wavefunction():
a = 1/Z
R = {
(1, 0): 2*sqrt(1/a**3) * exp(-r/a),
(2, 0): sqrt(1/(2*a**3)) * exp(-r/(2*a)) * (1 - r/(2*a)),
(2, 1): S.Half * sqrt(1/(6*a**3)) * exp(-r/(2*a)) * r/a,
(3, 0): Rational(2, 3) * sqrt(1/(3*a**3)) * exp(-r/(3*a)) *
(1 - 2*r/(3*a) + Rational(2, 27) * (r/a)**2),
(3, 1): Rational(4, 27) * sqrt(2/(3*a**3)) * exp(-r/(3*a)) *
(1 - r/(6*a)) * r/a,
(3, 2): Rational(2, 81) * sqrt(2/(15*a**3)) * exp(-r/(3*a)) * (r/a)**2,
(4, 0): Rational(1, 4) * sqrt(1/a**3) * exp(-r/(4*a)) *
(1 - 3*r/(4*a) + Rational(1, 8) * (r/a)**2 - Rational(1, 192) * (r/a)**3),
(4, 1): Rational(1, 16) * sqrt(5/(3*a**3)) * exp(-r/(4*a)) *
(1 - r/(4*a) + Rational(1, 80) * (r/a)**2) * (r/a),
(4, 2): Rational(1, 64) * sqrt(1/(5*a**3)) * exp(-r/(4*a)) *
(1 - r/(12*a)) * (r/a)**2,
(4, 3): Rational(1, 768) * sqrt(1/(35*a**3)) * exp(-r/(4*a)) * (r/a)**3,
}
for n, l in R:
assert simplify(R_nl(n, l, r, Z) - R[(n, l)]) == 0
def test_norm():
# Maximum "n" which is tested:
n_max = 2 # it works, but is slow, for n_max > 2
for n in range(n_max + 1):
for l in range(n):
assert integrate(R_nl(n, l, r)**2 * r**2, (r, 0, oo)) == 1
def test_psi_nlm():
r=S('r')
phi=S('phi')
theta=S('theta')
assert (Psi_nlm(1, 0, 0, r, phi, theta) == exp(-r) / sqrt(pi))
assert (Psi_nlm(2, 1, -1, r, phi, theta)) == S.Half * exp(-r / (2)) * r \
* (sin(theta) * exp(-I * phi) / (4 * sqrt(pi)))
assert (Psi_nlm(3, 2, 1, r, phi, theta, 2) == -sqrt(2) * sin(theta) \
* exp(I * phi) * cos(theta) / (4 * sqrt(pi)) * S(2) / 81 \
* sqrt(2 * 2 ** 3) * exp(-2 * r / (3)) * (r * 2) ** 2)
def test_hydrogen_energies():
assert E_nl(n, Z) == -Z**2/(2*n**2)
assert E_nl(n) == -1/(2*n**2)
assert E_nl(1, 47) == -S(47)**2/(2*1**2)
assert E_nl(2, 47) == -S(47)**2/(2*2**2)
assert E_nl(1) == -S.One/(2*1**2)
assert E_nl(2) == -S.One/(2*2**2)
assert E_nl(3) == -S.One/(2*3**2)
assert E_nl(4) == -S.One/(2*4**2)
assert E_nl(100) == -S.One/(2*100**2)
raises(ValueError, lambda: E_nl(0))
def test_hydrogen_energies_relat():
# First test exact formulas for small "c" so that we get nice expressions:
assert E_nl_dirac(2, 0, Z=1, c=1) == 1/sqrt(2) - 1
assert simplify(E_nl_dirac(2, 0, Z=1, c=2) - ( (8*sqrt(3) + 16)
/ sqrt(16*sqrt(3) + 32) - 4)) == 0
assert simplify(E_nl_dirac(2, 0, Z=1, c=3) - ( (54*sqrt(2) + 81)
/ sqrt(108*sqrt(2) + 162) - 9)) == 0
# Now test for almost the correct speed of light, without floating point
# numbers:
assert simplify(E_nl_dirac(2, 0, Z=1, c=137) - ( (352275361 + 10285412 *
sqrt(1173)) / sqrt(704550722 + 20570824 * sqrt(1173)) - 18769)) == 0
assert simplify(E_nl_dirac(2, 0, Z=82, c=137) - ( (352275361 + 2571353 *
sqrt(12045)) / sqrt(704550722 + 5142706*sqrt(12045)) - 18769)) == 0
# Test using exact speed of light, and compare against the nonrelativistic
# energies:
for n in range(1, 5):
for l in range(n):
assert feq(E_nl_dirac(n, l), E_nl(n), 1e-5, 1e-5)
if l > 0:
assert feq(E_nl_dirac(n, l, False), E_nl(n), 1e-5, 1e-5)
Z = 2
for n in range(1, 5):
for l in range(n):
assert feq(E_nl_dirac(n, l, Z=Z), E_nl(n, Z), 1e-4, 1e-4)
if l > 0:
assert feq(E_nl_dirac(n, l, False, Z), E_nl(n, Z), 1e-4, 1e-4)
Z = 3
for n in range(1, 5):
for l in range(n):
assert feq(E_nl_dirac(n, l, Z=Z), E_nl(n, Z), 1e-3, 1e-3)
if l > 0:
assert feq(E_nl_dirac(n, l, False, Z), E_nl(n, Z), 1e-3, 1e-3)
# Test the exceptions:
raises(ValueError, lambda: E_nl_dirac(0, 0))
raises(ValueError, lambda: E_nl_dirac(1, -1))
raises(ValueError, lambda: E_nl_dirac(1, 0, False))
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