def test_commutation():
n, m = symbols("n,m", above_fermi=True)
c = Commutator(B(0), Bd(0))
assert c == 1
c = Commutator(Bd(0), B(0))
assert c == -1
c = Commutator(B(n), Bd(0))
assert c == KroneckerDelta(n, 0)
c = Commutator(B(0), Bd(0))
e = simplify(apply_operators(c * BKet([n])))
assert e == BKet([n])
c = Commutator(B(0), B(1))
e = simplify(apply_operators(c * BKet([n, m])))
assert e == 0
c = Commutator(F(m), Fd(m))
assert c == +1 - 2 * NO(Fd(m) * F(m))
c = Commutator(Fd(m), F(m))
assert c.expand() == -1 + 2 * NO(Fd(m) * F(m))
C = Commutator
X, Y, Z = symbols('X,Y,Z', commutative=False)
assert C(C(X, Y), Z) != 0
assert C(C(X, Z), Y) != 0
assert C(Y, C(X, Z)) != 0
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
D = KroneckerDelta
assert C(Fd(a), F(i)) == -2 * NO(F(i) * Fd(a))
assert C(Fd(j), NO(Fd(a) * F(i))).doit(wicks=True) == -D(j, i) * Fd(a)
assert C(Fd(a) * F(i), Fd(b) * F(j)).doit(wicks=True) == 0
def test_create():
i, j, n, m = var('i j n m')
o = Bd(i)
assert isinstance(o, CreateBoson)
o = o.subs(i, j)
assert o.atoms(Symbol) == set([j])
o = Bd(0)
assert o.apply_operator(Ket([n])) == sqrt(n + 1) * Ket([n + 1])
o = Bd(n)
assert o.apply_operator(Ket([n])) == o * Ket([n])
def test_create_b():
i, j, n, m = symbols("i,j,n,m")
o = Bd(i)
assert isinstance(o, CreateBoson)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = Bd(0)
assert o.apply_operator(BKet([n])) == sqrt(n + 1) * BKet([n + 1])
o = Bd(n)
assert o.apply_operator(BKet([n])) == o * BKet([n])
def test_create():
i, j, n, m = symbols('i,j,n,m')
o = Bd(i)
assert latex(o) == "b^\\dagger_{i}"
assert isinstance(o, CreateBoson)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = Bd(0)
assert o.apply_operator(BKet([n])) == sqrt(n + 1)*BKet([n + 1])
o = Bd(n)
assert o.apply_operator(BKet([n])) == o*BKet([n])
def test_commutation():
n, m = symbols("n,m", above_fermi=True)
c = Commutator(B(0), Bd(0))
assert c == 1
c = Commutator(Bd(0), B(0))
assert c == -1
c = Commutator(B(n), Bd(0))
assert c == KroneckerDelta(n, 0)
c = Commutator(B(0), B(0))
assert c == 0
c = Commutator(B(0), Bd(0))
e = simplify(apply_operators(c * BKet([n])))
assert e == BKet([n])
c = Commutator(B(0), B(1))
e = simplify(apply_operators(c * BKet([n, m])))
assert e == 0
c = Commutator(F(m), Fd(m))
assert c == +1 - 2 * NO(Fd(m) * F(m))
c = Commutator(Fd(m), F(m))
assert c.expand() == -1 + 2 * NO(Fd(m) * F(m))
C = Commutator
X, Y, Z = symbols("X,Y,Z", commutative=False)
assert C(C(X, Y), Z) != 0
assert C(C(X, Z), Y) != 0
assert C(Y, C(X, Z)) != 0
i, j, k, l = symbols("i,j,k,l", below_fermi=True)
a, b, c, d = symbols("a,b,c,d", above_fermi=True)
p, q, r, s = symbols("p,q,r,s")
D = KroneckerDelta
assert C(Fd(a), F(i)) == -2 * NO(F(i) * Fd(a))
assert C(Fd(j), NO(Fd(a) * F(i))).doit(wicks=True) == -D(j, i) * Fd(a)
assert C(Fd(a) * F(i), Fd(b) * F(j)).doit(wicks=True) == 0
c1 = Commutator(F(a), Fd(a))
assert Commutator.eval(c1, c1) == 0
c = Commutator(Fd(a) * F(i), Fd(b) * F(j))
assert latex(c) == r"\left[a^\dagger_{a} a_{i},a^\dagger_{b} a_{j}\right]"
assert (
repr(c)
== "Commutator(CreateFermion(a)*AnnihilateFermion(i),CreateFermion(b)*AnnihilateFermion(j))"
)
assert (
str(c)
== "[CreateFermion(a)*AnnihilateFermion(i),CreateFermion(b)*AnnihilateFermion(j)]"
)
def test_commutation():
c = commutator(B(0), Bd(0))
e = simplify(apply_operators(c * Ket([n])))
assert e == Ket([n])
c = commutator(B(0), B(1))
e = simplify(apply_operators(c * Ket([n, m])))
assert e == 0
def test_matrix_elements():
b = VarBosonicBasis(5)
o = B(0)
m = matrix_rep(o, b)
for i in range(4):
assert m[i, i + 1] == sqrt(i + 1)
o = Bd(0)
m = matrix_rep(o, b)
for i in range(4):
assert m[i + 1, i] == sqrt(i + 1)
def test_sho():
n, m = symbols("n,m")
h_n = Bd(n) * B(n) * (n + S.Half)
H = Sum(h_n, (n, 0, 5))
o = H.doit(deep=False)
b = FixedBosonicBasis(2, 6)
m = matrix_rep(o, b)
# We need to double check these energy values to make sure that they
# are correct and have the proper degeneracies!
diag = [1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11]
for i in range(len(diag)):
assert diag[i] == m[i, i]
def test_create():
i, j, n, m = symbols('i j n m')
o = Bd(i)
assert isinstance(o, CreateBoson)
o = o.subs(i, j)
assert o.atoms(Symbol) == set([j])
o = Bd(0)
assert o.apply_operator(BKet([n])) == sqrt(n+1)*BKet([n+1])
o = Bd(n)
assert o.apply_operator(BKet([n])) == o*BKet([n])
def test_create():
i, j, n, m = symbols('i,j,n,m')
o = Bd(i)
assert latex(o) == "b^\\dagger_{i}"
assert isinstance(o, CreateBoson)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = Bd(0)
assert o.apply_operator(BKet([n])) == sqrt(n + 1)*BKet([n + 1])
o = Bd(n)
assert o.apply_operator(BKet([n])) == o*BKet([n])
def test_dagger():
i, j, n, m = symbols('i,j,n,m')
assert Dagger(1) == 1
assert Dagger(1.0) == 1.0
assert Dagger(2 * I) == -2 * I
assert Dagger(Rational(1, 2) * I / 3.0) == -Rational(1, 2) * I / 3.0
assert Dagger(BKet([n])) == BBra([n])
assert Dagger(B(0)) == Bd(0)
assert Dagger(Bd(0)) == B(0)
assert Dagger(B(n)) == Bd(n)
assert Dagger(Bd(n)) == B(n)
assert Dagger(B(0) + B(1)) == Bd(0) + Bd(1)
assert Dagger(n * m) == Dagger(n) * Dagger(m) # n, m commute
assert Dagger(B(n) * B(m)) == Bd(m) * Bd(n)
assert Dagger(B(n)**10) == Dagger(B(n))**10
def test_dagger():
i, j, n, m = symbols("i,j,n,m")
assert Dagger(1) == 1
assert Dagger(1.0) == 1.0
assert Dagger(2 * I) == -2 * I
assert Dagger(S.Half * I / 3.0) == I * Rational(-1, 2) / 3.0
assert Dagger(BKet([n])) == BBra([n])
assert Dagger(B(0)) == Bd(0)
assert Dagger(Bd(0)) == B(0)
assert Dagger(B(n)) == Bd(n)
assert Dagger(Bd(n)) == B(n)
assert Dagger(B(0) + B(1)) == Bd(0) + Bd(1)
assert Dagger(n * m) == Dagger(n) * Dagger(m) # n, m commute
assert Dagger(B(n) * B(m)) == Bd(m) * Bd(n)
assert Dagger(B(n) ** 10) == Dagger(B(n)) ** 10
assert Dagger("a") == Dagger(Symbol("a"))
assert Dagger(Dagger("a")) == Symbol("a")
def test_NO():
i, j, k, l = symbols("i j k l", below_fermi=True)
a, b, c, d = symbols("a b c d", above_fermi=True)
p, q, r, s = symbols("p q r s", cls=Dummy)
assert NO(Fd(p) * F(q) + Fd(a) * F(b)) == NO(Fd(p) * F(q)) + NO(Fd(a) * F(b))
assert NO(Fd(i) * NO(F(j) * Fd(a))) == NO(Fd(i) * F(j) * Fd(a))
assert NO(1) == 1
assert NO(i) == i
assert NO(Fd(a) * Fd(b) * (F(c) + F(d))) == NO(Fd(a) * Fd(b) * F(c)) + NO(
Fd(a) * Fd(b) * F(d)
)
assert NO(Fd(a) * F(b))._remove_brackets() == Fd(a) * F(b)
assert NO(F(j) * Fd(i))._remove_brackets() == F(j) * Fd(i)
assert NO(Fd(p) * F(q)).subs(Fd(p), Fd(a) + Fd(i)) == NO(Fd(a) * F(q)) + NO(
Fd(i) * F(q)
)
assert NO(Fd(p) * F(q)).subs(F(q), F(a) + F(i)) == NO(Fd(p) * F(a)) + NO(
Fd(p) * F(i)
)
expr = NO(Fd(p) * F(q))._remove_brackets()
assert wicks(expr) == NO(expr)
assert NO(Fd(a) * F(b)) == -NO(F(b) * Fd(a))
no = NO(Fd(a) * F(i) * F(b) * Fd(j))
l1 = [ind for ind in no.iter_q_creators()]
assert l1 == [0, 1]
l2 = [ind for ind in no.iter_q_annihilators()]
assert l2 == [3, 2]
no = NO(Fd(a) * Fd(i))
assert no.has_q_creators == 1
assert no.has_q_annihilators == -1
assert str(no) == ":CreateFermion(a)*CreateFermion(i):"
assert repr(no) == "NO(CreateFermion(a)*CreateFermion(i))"
assert latex(no) == r"\left\{a^\dagger_{a} a^\dagger_{i}\right\}"
raises(NotImplementedError, lambda: NO(Bd(p) * F(q)))
def test_issue_19661():
a = Symbol('0')
assert latex(Commutator(
Bd(a)**2, B(a))) == '- \\left[b_{0},{b^\\dagger_{0}}^{2}\\right]'
def test_number_operator():
o = Bd(0) * B(0)
e = apply_operators(o * Ket([n]))
assert e == n * Ket([n])
def test_basic_apply():
e = B(0) * Ket([n])
assert apply_operators(e) == sqrt(n) * Ket([n - 1])
e = Bd(0) * Ket([n])
assert apply_operators(e) == sqrt(n + 1) * Ket([n + 1])
def test_basic_apply():
n = symbols("n")
e = B(0) * BKet([n])
assert apply_operators(e) == sqrt(n) * BKet([n - 1])
e = Bd(0) * BKet([n])
assert apply_operators(e) == sqrt(n + 1) * BKet([n + 1])
def test_complex_apply():
n, m = symbols("n,m")
o = Bd(0) * B(0) * Bd(1) * B(0)
e = apply_operators(o * BKet([n, m]))
answer = sqrt(n) * sqrt(m + 1) * (-1 + n) * BKet([-1 + n, 1 + m])
assert expand(e) == expand(answer)
def test_number_operator():
n = symbols("n")
o = Bd(0) * B(0)
e = apply_operators(o * BKet([n]))
assert e == n * BKet([n])