def test_wicks():
p, q, r, s = symbols("p,q,r,s", above_fermi=True)
# Testing for particles only
str = F(p) * Fd(q)
assert wicks(str) == NO(F(p) * Fd(q)) + KroneckerDelta(p, q)
str = Fd(p) * F(q)
assert wicks(str) == NO(Fd(p) * F(q))
str = F(p) * Fd(q) * F(r) * Fd(s)
nstr = wicks(str)
fasit = NO(
KroneckerDelta(p, q) * KroneckerDelta(r, s)
+ KroneckerDelta(p, q) * AnnihilateFermion(r) * CreateFermion(s)
+ KroneckerDelta(r, s) * AnnihilateFermion(p) * CreateFermion(q)
- KroneckerDelta(p, s) * AnnihilateFermion(r) * CreateFermion(q)
- AnnihilateFermion(p)
* AnnihilateFermion(r)
* CreateFermion(q)
* CreateFermion(s)
)
assert nstr == fasit
assert (p * q * nstr).expand() == wicks(p * q * str)
assert (nstr * p * q * 2).expand() == wicks(str * p * q * 2)
# Testing CC equations particles and holes
i, j, k, l = symbols("i j k l", below_fermi=True, cls=Dummy)
a, b, c, d = symbols("a b c d", above_fermi=True, cls=Dummy)
p, q, r, s = symbols("p q r s", cls=Dummy)
assert wicks(F(a) * NO(F(i) * F(j)) * Fd(b)) == NO(
F(a) * F(i) * F(j) * Fd(b)
) + KroneckerDelta(a, b) * NO(F(i) * F(j))
assert wicks(F(a) * NO(F(i) * F(j) * F(k)) * Fd(b)) == NO(
F(a) * F(i) * F(j) * F(k) * Fd(b)
) - KroneckerDelta(a, b) * NO(F(i) * F(j) * F(k))
expr = wicks(Fd(i) * NO(Fd(j) * F(k)) * F(l))
assert expr == -KroneckerDelta(i, k) * NO(Fd(j) * F(l)) - KroneckerDelta(j, l) * NO(
Fd(i) * F(k)
) - KroneckerDelta(i, k) * KroneckerDelta(j, l) + KroneckerDelta(i, l) * NO(
Fd(j) * F(k)
) + NO(
Fd(i) * Fd(j) * F(k) * F(l)
)
expr = wicks(F(a) * NO(F(b) * Fd(c)) * Fd(d))
assert expr == -KroneckerDelta(a, c) * NO(F(b) * Fd(d)) - KroneckerDelta(b, d) * NO(
F(a) * Fd(c)
) - KroneckerDelta(a, c) * KroneckerDelta(b, d) + KroneckerDelta(a, d) * NO(
F(b) * Fd(c)
) + NO(
F(a) * F(b) * Fd(c) * Fd(d)
)
def test_sympy__physics__secondquant__AnnihilateFermion():
from sympy.physics.secondquant import AnnihilateFermion
assert _test_args(AnnihilateFermion(0))