def test_x(): assert X.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity)) assert Commutator(X, Px).doit() == I * hbar assert qapply(X * XKet(x)) == x * XKet(x) assert XKet(x).dual_class() == XBra assert XBra(x).dual_class() == XKet assert (Dagger(XKet(y)) * XKet(x)).doit() == DiracDelta(x - y) assert (PxBra(px)*XKet(x)).doit() == \ exp(-I*x*px/hbar)/sqrt(2*pi*hbar) assert represent(XKet(x)) == DiracDelta(x - x_1) assert represent(XBra(x)) == DiracDelta(-x + x_1) assert XBra(x).position == x assert represent(XOp() * XKet()) == x * DiracDelta(x - x_2) assert represent(XOp()*XKet()*XBra('y')) == \ x*DiracDelta(x - x_3)*DiracDelta(x_1 - y) assert represent(XBra("y") * XKet()) == DiracDelta(x - y) assert represent(XKet() * XBra()) == DiracDelta(x - x_2) * DiracDelta(x_1 - x) rep_p = represent(XOp(), basis=PxOp) assert rep_p == hbar * I * DiracDelta(px_1 - px_2) * DifferentialOperator(px_1) assert rep_p == represent(XOp(), basis=PxOp()) assert rep_p == represent(XOp(), basis=PxKet) assert rep_p == represent(XOp(), basis=PxKet()) assert represent(XOp()*PxKet(), basis=PxKet) == \ hbar*I*DiracDelta(px - px_2)*DifferentialOperator(px)
def test_operator(): a = Operator('A') b = Operator('B', Symbol('t'), S(1) / 2) inv = a.inv() f = Function('f') x = symbols('x') d = DifferentialOperator(Derivative(f(x), x), f(x)) op = OuterProduct(Ket(), Bra()) assert str(a) == 'A' assert pretty(a) == 'A' assert upretty(a) == u'A' assert latex(a) == 'A' sT(a, "Operator(Symbol('A'))") assert str(inv) == 'A**(-1)' ascii_str = \ """\ -1\n\ A \ """ ucode_str = \ u"""\ -1\n\ A \ """ assert pretty(inv) == ascii_str assert upretty(inv) == ucode_str assert latex(inv) == r'\left(A\right)^{-1}' sT(inv, "Pow(Operator(Symbol('A')), Integer(-1))") assert str(d) == 'DifferentialOperator(Derivative(f(x), x),f(x))' ascii_str = \ """\ /d \\\n\ DifferentialOperator|--(f(x)),f(x)|\n\ \dx /\ """ ucode_str = \ u"""\ ⎛d ⎞\n\ DifferentialOperator⎜──(f(x)),f(x)⎟\n\ ⎝dx ⎠\ """ assert pretty(d) == ascii_str assert upretty(d) == ucode_str assert latex(d) == \ r'DifferentialOperator\left(\frac{\partial}{\partial x} \operatorname{f}{\left (x \right )},\operatorname{f}{\left (x \right )}\right)' sT( d, "DifferentialOperator(Derivative(Function('f')(Symbol('x')), Symbol('x')),Function('f')(Symbol('x')))" ) assert str(b) == 'Operator(B,t,1/2)' assert pretty(b) == 'Operator(B,t,1/2)' assert upretty(b) == u'Operator(B,t,1/2)' assert latex(b) == r'Operator\left(B,t,\frac{1}{2}\right)' sT(b, "Operator(Symbol('B'),Symbol('t'),Rational(1, 2))") assert str(op) == '|psi><psi|' assert pretty(op) == '|psi><psi|' assert upretty(op) == u'❘ψ⟩⟨ψ❘' assert latex(op) == r'{\left|\psi\right\rangle }{\left\langle \psi\right|}' sT(op, "OuterProduct(Ket(Symbol('psi')),Bra(Symbol('psi')))")
def _represent_PxKet(self, basis, *, index=1, **options): states = basis._enumerate_state(2, start_index=index) coord1 = states[0].momentum coord2 = states[1].momentum d = DifferentialOperator(coord1) delta = DiracDelta(coord1 - coord2) return I * hbar * (d * delta)
def test_operator(): a = Operator("A") b = Operator("B", Symbol("t"), S.Half) inv = a.inv() f = Function("f") x = symbols("x") d = DifferentialOperator(Derivative(f(x), x), f(x)) op = OuterProduct(Ket(), Bra()) assert str(a) == "A" assert pretty(a) == "A" assert upretty(a) == u"A" assert latex(a) == "A" sT(a, "Operator(Symbol('A'))") assert str(inv) == "A**(-1)" ascii_str = """\ -1\n\ A \ """ ucode_str = u("""\ -1\n\ A \ """) assert pretty(inv) == ascii_str assert upretty(inv) == ucode_str assert latex(inv) == r"A^{-1}" sT(inv, "Pow(Operator(Symbol('A')), Integer(-1))") assert str(d) == "DifferentialOperator(Derivative(f(x), x),f(x))" ascii_str = """\ /d \\\n\ DifferentialOperator|--(f(x)),f(x)|\n\ \\dx /\ """ ucode_str = u("""\ ⎛d ⎞\n\ DifferentialOperator⎜──(f(x)),f(x)⎟\n\ ⎝dx ⎠\ """) assert pretty(d) == ascii_str assert upretty(d) == ucode_str assert ( latex(d) == r"DifferentialOperator\left(\frac{d}{d x} f{\left(x \right)},f{\left(x \right)}\right)" ) sT( d, "DifferentialOperator(Derivative(Function('f')(Symbol('x')), Tuple(Symbol('x'), Integer(1))),Function('f')(Symbol('x')))", ) assert str(b) == "Operator(B,t,1/2)" assert pretty(b) == "Operator(B,t,1/2)" assert upretty(b) == u"Operator(B,t,1/2)" assert latex(b) == r"Operator\left(B,t,\frac{1}{2}\right)" sT(b, "Operator(Symbol('B'),Symbol('t'),Rational(1, 2))") assert str(op) == "|psi><psi|" assert pretty(op) == "|psi><psi|" assert upretty(op) == u"❘ψ⟩⟨ψ❘" assert latex(op) == r"{\left|\psi\right\rangle }{\left\langle \psi\right|}" sT(op, "OuterProduct(Ket(Symbol('psi')),Bra(Symbol('psi')))")
def _represent_XKet(self, basis, **options): index = options.pop("index", 1) states = basis._enumerate_state(2, start_index=index) coord1 = states[0].position coord2 = states[1].position d = DifferentialOperator(coord1) delta = DiracDelta(coord1 - coord2) return -I * hbar * (d * delta)
def test_p(): assert Px.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity)) assert qapply(Px*PxKet(px)) == px*PxKet(px) assert PxKet(px).dual_class() == PxBra assert PxBra(x).dual_class() == PxKet assert (Dagger(PxKet(py))*PxKet(px)).doit() == DiracDelta(px-py) assert (XBra(x)*PxKet(px)).doit() ==\ exp(I*x*px/hbar)/sqrt(2*pi*hbar) assert represent(PxKet(px)) == DiracDelta(px-px_1) rep_x = represent(PxOp(), basis = XOp) assert rep_x == -hbar*I*DiracDelta(x_1 - x_2)*DifferentialOperator(x_1) assert rep_x == represent(PxOp(), basis = XOp()) assert rep_x == represent(PxOp(), basis = XKet) assert rep_x == represent(PxOp(), basis = XKet()) assert represent(PxOp()*XKet(), basis=XKet) == \ -hbar*I*DiracDelta(x - x_2)*DifferentialOperator(x) assert represent(XBra("y")*PxOp()*XKet(), basis=XKet) == \ -hbar*I*DiracDelta(x-y)*DifferentialOperator(x)
def test_sympy__physics__quantum__operator__DifferentialOperator(): from sympy.physics.quantum.operator import DifferentialOperator from sympy import Derivative, Function f = Function('f') assert _test_args(DifferentialOperator(1 / x * Derivative(f(x), x), f(x)))
def test_differential_operator(): x = Symbol('x') f = Function('f') d = DifferentialOperator(Derivative(f(x), x), f(x)) g = Wavefunction(x**2, x) assert qapply(d*g) == Wavefunction(2*x, x) assert d.expr == Derivative(f(x), x) assert d.function == f(x) assert d.variables == (x,) assert diff(d, x) == DifferentialOperator(Derivative(f(x), x, 2), f(x)) d = DifferentialOperator(Derivative(f(x), x, 2), f(x)) g = Wavefunction(x**3, x) assert qapply(d*g) == Wavefunction(6*x, x) assert d.expr == Derivative(f(x), x, 2) assert d.function == f(x) assert d.variables == (x,) assert diff(d, x) == DifferentialOperator(Derivative(f(x), x, 3), f(x)) d = DifferentialOperator(1/x*Derivative(f(x), x), f(x)) assert d.expr == 1/x*Derivative(f(x), x) assert d.function == f(x) assert d.variables == (x,) assert diff(d, x) == \ DifferentialOperator(Derivative(1/x*Derivative(f(x), x), x), f(x)) assert qapply(d*g) == Wavefunction(3*x, x) # 2D cartesian Laplacian y = Symbol('y') d = DifferentialOperator(Derivative(f(x, y), x, 2) + Derivative(f(x, y), y, 2), f(x, y)) w = Wavefunction(x**3*y**2 + y**3*x**2, x, y) assert d.expr == Derivative(f(x, y), x, 2) + Derivative(f(x, y), y, 2) assert d.function == f(x, y) assert d.variables == (x, y) assert diff(d, x) == \ DifferentialOperator(Derivative(d.expr, x), f(x, y)) assert diff(d, y) == \ DifferentialOperator(Derivative(d.expr, y), f(x, y)) assert qapply(d*w) == Wavefunction(2*x**3 + 6*x*y**2 + 6*x**2*y + 2*y**3, x, y) # 2D polar Laplacian (th = theta) r, th = symbols('r th') d = DifferentialOperator(1/r*Derivative(r*Derivative(f(r, th), r), r) + 1/(r**2)*Derivative(f(r, th), th, 2), f(r, th)) w = Wavefunction(r**2*sin(th), r, (th, 0, pi)) assert d.expr == \ 1/r*Derivative(r*Derivative(f(r, th), r), r) + \ 1/(r**2)*Derivative(f(r, th), th, 2) assert d.function == f(r, th) assert d.variables == (r, th) assert diff(d, r) == \ DifferentialOperator(Derivative(d.expr, r), f(r, th)) assert diff(d, th) == \ DifferentialOperator(Derivative(d.expr, th), f(r, th)) assert qapply(d*w) == Wavefunction(3*sin(th), r, (th, 0, pi))
def test_big_expr(): f = Function('f') x = symbols('x') e1 = Dagger( AntiCommutator( Operator('A') + Operator('B'), Pow(DifferentialOperator(Derivative(f(x), x), f(x)), 3)) * TensorProduct(Jz**2, Operator('A') + Operator('B'))) * (JzBra(1, 0) + JzBra( 1, 1)) * (JzKet(0, 0) + JzKet(1, -1)) e2 = Commutator(Jz**2, Operator('A') + Operator('B')) * AntiCommutator( Dagger(Operator('C') * Operator('D')), Operator('E').inv()**2) * Dagger(Commutator(Jz, J2)) e3 = Wigner3j(1, 2, 3, 4, 5, 6) * TensorProduct( Commutator( Operator('A') + Dagger(Operator('B')), Operator('C') + Operator('D')), Jz - J2) * Dagger( OuterProduct(Dagger(JzBra(1, 1)), JzBra( 1, 0))) * TensorProduct( JzKetCoupled(1, 1, (1, 1)) + JzKetCoupled(1, 0, (1, 1)), JzKetCoupled(1, -1, (1, 1))) e4 = (ComplexSpace(1) * ComplexSpace(2) + FockSpace()**2) * (L2(Interval(0, oo)) + HilbertSpace()) assert str( e1 ) == '(Jz**2)x(Dagger(A) + Dagger(B))*{Dagger(DifferentialOperator(Derivative(f(x), x),f(x)))**3,Dagger(A) + Dagger(B)}*(<1,0| + <1,1|)*(|0,0> + |1,-1>)' ascii_str = \ """\ / 3 \\ \n\ |/ +\\ | \n\ 2 / + +\\ <| /d \\ | + +> \n\ /J \\ x \\A + B /*||DifferentialOperator|--(f(x)),f(x)| | ,A + B |*(<1,0| + <1,1|)*(|0,0> + |1,-1>)\n\ \\ z/ \\\\ \dx / / / \ """ ucode_str = \ u"""\ ⎧ 3 ⎫ \n\ ⎪⎛ †⎞ ⎪ \n\ 2 ⎛ † †⎞ ⎨⎜ ⎛d ⎞ ⎟ † †⎬ \n\ ⎛J ⎞ ⨂ ⎝A + B ⎠⋅⎪⎜DifferentialOperator⎜──(f(x)),f(x)⎟ ⎟ ,A + B ⎪⋅(⟨1,0❘ + ⟨1,1❘)⋅(❘0,0⟩ + ❘1,-1⟩)\n\ ⎝ z⎠ ⎩⎝ ⎝dx ⎠ ⎠ ⎭ \ """ assert pretty(e1) == ascii_str assert upretty(e1) == ucode_str assert latex(e1) == \ r'{\left(J_z\right)^{2}}\otimes \left({A^{\dag} + B^{\dag}}\right) \left\{\left(DifferentialOperator\left(\frac{\partial}{\partial x} \operatorname{f}{\left (x \right )},\operatorname{f}{\left (x \right )}\right)^{\dag}\right)^{3},A^{\dag} + B^{\dag}\right\} \left({\left\langle 1,0\right|} + {\left\langle 1,1\right|}\right) \left({\left|0,0\right\rangle } + {\left|1,-1\right\rangle }\right)' sT( e1, "Mul(TensorProduct(Pow(JzOp(Symbol('J')), Integer(2)), Add(Dagger(Operator(Symbol('A'))), Dagger(Operator(Symbol('B'))))), AntiCommutator(Pow(Dagger(DifferentialOperator(Derivative(Function('f')(Symbol('x')), Symbol('x')),Function('f')(Symbol('x')))), Integer(3)),Add(Dagger(Operator(Symbol('A'))), Dagger(Operator(Symbol('B'))))), Add(JzBra(Integer(1),Integer(0)), JzBra(Integer(1),Integer(1))), Add(JzKet(Integer(0),Integer(0)), JzKet(Integer(1),Integer(-1))))" ) assert str(e2) == '[Jz**2,A + B]*{E**(-2),Dagger(D)*Dagger(C)}*[J2,Jz]' ascii_str = \ """\ [ 2 ] / -2 + +\\ [ 2 ]\n\ [/J \\ ,A + B]*<E ,D *C >*[J ,J ]\n\ [\\ z/ ] \\ / [ z]\ """ ucode_str = \ u"""\ ⎡ 2 ⎤ ⎧ -2 † †⎫ ⎡ 2 ⎤\n\ ⎢⎛J ⎞ ,A + B⎥⋅⎨E ,D ⋅C ⎬⋅⎢J ,J ⎥\n\ ⎣⎝ z⎠ ⎦ ⎩ ⎭ ⎣ z⎦\ """ assert pretty(e2) == ascii_str assert upretty(e2) == ucode_str assert latex(e2) == \ r'\left[\left(J_z\right)^{2},A + B\right] \left\{\left(E\right)^{-2},D^{\dag} C^{\dag}\right\} \left[J^2,J_z\right]' sT( e2, "Mul(Commutator(Pow(JzOp(Symbol('J')), Integer(2)),Add(Operator(Symbol('A')), Operator(Symbol('B')))), AntiCommutator(Pow(Operator(Symbol('E')), Integer(-2)),Mul(Dagger(Operator(Symbol('D'))), Dagger(Operator(Symbol('C'))))), Commutator(J2Op(Symbol('J')),JzOp(Symbol('J'))))" ) assert str(e3) == \ "Wigner3j(1, 2, 3, 4, 5, 6)*[Dagger(B) + A,C + D]x(-J2 + Jz)*|1,0><1,1|*(|1,0,j1=1,j2=1> + |1,1,j1=1,j2=1>)x|1,-1,j1=1,j2=1>" ascii_str = \ """\ [ + ] / 2 \\ \n\ /1 3 5\\*[B + A,C + D]x |- J + J |*|1,0><1,1|*(|1,0,j1=1,j2=1> + |1,1,j1=1,j2=1>)x |1,-1,j1=1,j2=1>\n\ | | \\ z/ \n\ \\2 4 6/ \ """ ucode_str = \ u"""\ ⎡ † ⎤ ⎛ 2 ⎞ \n\ ⎛1 3 5⎞⋅⎣B + A,C + D⎦⨂ ⎜- J + J ⎟⋅❘1,0⟩⟨1,1❘⋅(❘1,0,j₁=1,j₂=1⟩ + ❘1,1,j₁=1,j₂=1⟩)⨂ ❘1,-1,j₁=1,j₂=1⟩\n\ ⎜ ⎟ ⎝ z⎠ \n\ ⎝2 4 6⎠ \ """ assert pretty(e3) == ascii_str assert upretty(e3) == ucode_str assert latex(e3) == \ r'\left(\begin{array}{ccc} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array}\right) {\left[B^{\dag} + A,C + D\right]}\otimes \left({- J^2 + J_z}\right) {\left|1,0\right\rangle }{\left\langle 1,1\right|} \left({{\left|1,0,j_{1}=1,j_{2}=1\right\rangle } + {\left|1,1,j_{1}=1,j_{2}=1\right\rangle }}\right)\otimes {{\left|1,-1,j_{1}=1,j_{2}=1\right\rangle }}' sT( e3, "Mul(Wigner3j(Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6)), TensorProduct(Commutator(Add(Dagger(Operator(Symbol('B'))), Operator(Symbol('A'))),Add(Operator(Symbol('C')), Operator(Symbol('D')))), Add(Mul(Integer(-1), J2Op(Symbol('J'))), JzOp(Symbol('J')))), OuterProduct(JzKet(Integer(1),Integer(0)),JzBra(Integer(1),Integer(1))), TensorProduct(Add(JzKetCoupled(Integer(1),Integer(0),Tuple(Integer(1), Integer(1)),Tuple(Tuple(Integer(1), Integer(2), Integer(1)))), JzKetCoupled(Integer(1),Integer(1),Tuple(Integer(1), Integer(1)),Tuple(Tuple(Integer(1), Integer(2), Integer(1))))), JzKetCoupled(Integer(1),Integer(-1),Tuple(Integer(1), Integer(1)),Tuple(Tuple(Integer(1), Integer(2), Integer(1))))))" ) assert str(e4) == '(C(1)*C(2)+F**2)*(L2([0, oo))+H)' ascii_str = \ """\ // 1 2\\ x2\\ / 2 \\\n\ \\\\C x C / + F / x \L + H/\ """ ucode_str = \ u"""\ ⎛⎛ 1 2⎞ ⨂2⎞ ⎛ 2 ⎞\n\ ⎝⎝C ⨂ C ⎠ ⊕ F ⎠ ⨂ ⎝L ⊕ H⎠\ """ assert pretty(e4) == ascii_str assert upretty(e4) == ucode_str assert latex(e4) == \ r'\left(\left(\mathcal{C}^{1}\otimes \mathcal{C}^{2}\right)\oplus {\mathcal{F}}^{\otimes 2}\right)\otimes \left({\mathcal{L}^2}\left( \left[0, \infty\right) \right)\oplus \mathcal{H}\right)' sT( e4, "TensorProductHilbertSpace((DirectSumHilbertSpace(TensorProductHilbertSpace(ComplexSpace(Integer(1)),ComplexSpace(Integer(2))),TensorPowerHilbertSpace(FockSpace(),Integer(2)))),(DirectSumHilbertSpace(L2(Interval(Integer(0), oo, False, True)),HilbertSpace())))" )
def test_operator(): a = Operator('A') b = Operator('B', Symbol('t'), S(1) / 2) inv = a.inv() f = Function('f') x = symbols('x') d = DifferentialOperator(Derivative(f(x), x), f(x)) op = OuterProduct(Ket(), Bra()) assert str(a) == 'A' assert pretty(a) == 'A' assert upretty(a) == 'A' assert latex(a) == 'A' sT(a, "Operator(Symbol('A'))") assert str(inv) == 'A**(-1)' ascii_str = \ """\ -1\n\ A \ """ ucode_str = \ """\ -1\n\ A \ """ assert pretty(inv) == ascii_str assert upretty(inv) == ucode_str #FIXME ajgpitch 2019-09-22 # It's not clear to me why these extra brackets would be wanted / needed #assert latex(inv) == r'\left(A\right)^{-1}' # This renders okay assert latex(inv) == r'A^{-1}' sT(inv, "Pow(Operator(Symbol('A')), Integer(-1))") assert str(d) == 'DifferentialOperator(Derivative(f(x), x),f(x))' ascii_str = \ """\ /d \\\n\ DifferentialOperator|--(f(x)),f(x)|\n\ \dx /\ """ ucode_str = \ """\ ⎛d ⎞\n\ DifferentialOperator⎜──(f(x)),f(x)⎟\n\ ⎝dx ⎠\ """ assert pretty(d) == ascii_str assert upretty(d) == ucode_str assert latex(d) == \ r'DifferentialOperator\left(\frac{d}{d x} f{\left(x \right)},f{\left(x \right)}\right)' #FIXME: ajgpitch 2019-09-22 # Not clear why this is failing # `Tuple(Symbol('x'), Integer(1))` seems to enter into srepr(expr) # for some reason. sT( d, "DifferentialOperator(Derivative(Function('f')(Symbol('x')), Symbol('x')),Function('f')(Symbol('x')))" ) assert str(b) == 'Operator(B,t,1/2)' assert pretty(b) == 'Operator(B,t,1/2)' assert upretty(b) == 'Operator(B,t,1/2)' assert latex(b) == r'Operator\left(B,t,\frac{1}{2}\right)' sT(b, "Operator(Symbol('B'),Symbol('t'),Rational(1, 2))") assert str(op) == '|psi><psi|' assert pretty(op) == '|psi><psi|' assert upretty(op) == '❘ψ⟩⟨ψ❘' assert latex(op) == r'{\left|\psi\right\rangle }{\left\langle \psi\right|}' sT(op, "OuterProduct(Ket(Symbol('psi')),Bra(Symbol('psi')))")
import sympy as sym from sympy.physics.quantum import Operator from sympy.physics.quantum.operator import DifferentialOperator D0, D1, D2 = sym.symbols('D0 D1 D2', real=True) x, x0, x1, x2, x3 = sym.symbols('x x0 x1 x2 x3', real=True) eps, omega0, mu = sym.symbols('eps omega0 mu', real=True) A, B = sym.symbols('A B', real=True) T0, T1, T2 = sym.symbols('T0 T1 T2', real=True) D0 = Operator('D0') D1 = Operator('D1') f = sym.Function('f') d = DifferentialOperator(sym.diff(f(x), x), f(x)) print(sym.pretty(d(1 / x))) # ddt2 = D0**2 + 2 * eps * D0 * D1 + eps**2 * (D1**2 + 2 * D0 * D2) # ddt = D0 + eps * D1 ddt2 = D0**2 + 2 * eps * D0 * D1 ddt = D0 + eps * D1 # x = x0 + eps * x1 + eps**2 * x2 + eps**3 * x3 x = x0 + eps * x1 f = ddt * x - (ddt * x)**3 equ = ddt2 * x + omega0**2 * x - eps * f # equ = ddt2 * x + 2 * eps * mu * ddt * x + x + eps * x**3 equ = equ.expand() print(sym.pretty(equ.subs({eps: 0}))) print(sym.pretty(equ.coeff(eps))) # print(sym.pretty(equ.coeff(eps**2)))