def print_exact_energies(N, l, Z):
"""
Prints lowest N exact energies for the given "l" and "Z".
"""
print "Z=%d" % Z
print " n l Schroed Dirac (up) Dirac (down)"
print "-"*55
for n in range(l+1, l+N+1):
print "%2d %d %15.8f %15.8f" % (n, l, E_nl(n, Z=Z).n(),
E_nl_dirac(n, l, Z=Z)),
if l > 0:
print "%15.8f" % E_nl_dirac(n, l, False, Z=Z)
else:
print
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))
from sympy.physics.hydrogen import E_nl, E_nl_dirac, R_nl
from sympy import var
var("n Z")
var("r Z")
var("n l")
E_nl(n, Z)
E_nl(1)
E_nl(2, 4)
E_nl(n, l)
E_nl_dirac(5, 2) # l should be less than n
E_nl_dirac(2, 1)
E_nl_dirac(3, 2, False)
R_nl(5, 0, r) # z = 1 by default
R_nl(5, 0, r, 1)