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_sympy__physics__quantum__state__Wavefunction():
from sympy.physics.quantum.state import Wavefunction
from sympy.functions import sin
from sympy import Piecewise, pi
n = 1
L = 1
g = Piecewise((0, x < 0), (0, x > L),
(sqrt(2 // L) * sin(n * pi * x / L), True))
assert _test_args(Wavefunction(g, x))
def _apply_operator_Wavefunction(self, func):
from sympy.physics.quantum.state import Wavefunction
var = self.variables
wf_vars = func.args[1:]
f = self.function
new_expr = self.expr.subs(f, func(*var))
new_expr = new_expr.doit()
return Wavefunction(new_expr, *wf_vars)
def test_wavefunction():
x, y, L = symbols('x,y,L', real=True)
n = symbols('n', integer=True)
f = Wavefunction(x**2, x)
p = f.prob()
lims = f.limits
assert f.is_normalized is False
assert f.norm == oo
assert f(10) == 100
assert p(10) == 10000
assert lims[x] == (-oo, oo)
assert diff(f, x) == Wavefunction(2*x, x)
raises(NotImplementedError, lambda: f.normalize())
assert conjugate(f) == Wavefunction(conjugate(f.expr), x)
assert conjugate(f) == Dagger(f)
g = Wavefunction(x**2*y + y**2*x, (x, 0, 1), (y, 0, 2))
lims_g = g.limits
assert lims_g[x] == (0, 1)
assert lims_g[y] == (0, 2)
assert g.is_normalized is False
assert g.norm == sqrt(42)/3
assert g(2, 4) == 0
assert g(1, 1) == 2
assert diff(diff(g, x), y) == Wavefunction(2*x + 2*y, (x, 0, 1), (y, 0, 2))
assert conjugate(g) == Wavefunction(conjugate(g.expr), *g.args[1:])
assert conjugate(g) == Dagger(g)
h = Wavefunction(sqrt(5)*x**2, (x, 0, 1))
assert h.is_normalized is True
assert h.normalize() == h
assert conjugate(h) == Wavefunction(conjugate(h.expr), (x, 0, 1))
assert conjugate(h) == Dagger(h)
piab = Wavefunction(sin(n*pi*x/L), (x, 0, L))
assert piab.norm == sqrt(L/2)
assert piab(L + 1) == 0
assert piab(0.5) == sin(0.5*n*pi/L)
assert piab(0.5, n=1, L=1) == sin(0.5*pi)
assert piab.normalize() == \
Wavefunction(sqrt(2)/sqrt(L)*sin(n*pi*x/L), (x, 0, L))
assert conjugate(piab) == Wavefunction(conjugate(piab.expr), (x, 0, L))
assert conjugate(piab) == Dagger(piab)
k = Wavefunction(x**2, 'x')
assert type(k.variables[0]) == Symbol
def test_wavefunction():
x, y = symbols('x y', real=True)
L = symbols('L', positive=True)
n = symbols('n', integer=True, positive=True)
f = Wavefunction(x**2, x)
p = f.prob()
lims = f.limits
assert f.is_normalized is False
assert f.norm is oo
assert f(10) == 100
assert p(10) == 10000
assert lims[x] == (-oo, oo)
assert diff(f, x) == Wavefunction(2 * x, x)
raises(NotImplementedError, lambda: f.normalize())
assert conjugate(f) == Wavefunction(conjugate(f.expr), x)
assert conjugate(f) == Dagger(f)
g = Wavefunction(x**2 * y + y**2 * x, (x, 0, 1), (y, 0, 2))
lims_g = g.limits
assert lims_g[x] == (0, 1)
assert lims_g[y] == (0, 2)
assert g.is_normalized is False
assert g.norm == sqrt(42) / 3
assert g(2, 4) == 0
assert g(1, 1) == 2
assert diff(diff(g, x), y) == Wavefunction(2 * x + 2 * y, (x, 0, 1),
(y, 0, 2))
assert conjugate(g) == Wavefunction(conjugate(g.expr), *g.args[1:])
assert conjugate(g) == Dagger(g)
h = Wavefunction(sqrt(5) * x**2, (x, 0, 1))
assert h.is_normalized is True
assert h.normalize() == h
assert conjugate(h) == Wavefunction(conjugate(h.expr), (x, 0, 1))
assert conjugate(h) == Dagger(h)
piab = Wavefunction(sin(n * pi * x / L), (x, 0, L))
assert piab.norm == sqrt(L / 2)
assert piab(L + 1) == 0
assert piab(0.5) == sin(0.5 * n * pi / L)
assert piab(0.5, n=1, L=1) == sin(0.5 * pi)
assert piab.normalize() == \
Wavefunction(sqrt(2)/sqrt(L)*sin(n*pi*x/L), (x, 0, L))
assert conjugate(piab) == Wavefunction(conjugate(piab.expr), (x, 0, L))
assert conjugate(piab) == Dagger(piab)
k = Wavefunction(x**2, 'x')
assert type(k.variables[0]) == Symbol