def main():
# make environment
pot_region = (2/np.sqrt(3),2/np.sqrt(3),2/np.sqrt(3))
radius = np.sqrt(pot_region[0] **2 + pot_region[1] **2 + pot_region[2] **2 )
region = (radius,radius,radius)
nr = 201
gridpx = 200
gridpy = 200
gridpz = 200
x,y,z = grid(gridpx,gridpy,gridpz,region)
xx, yy, zz = np.meshgrid(x,y,z)
#make mesh
#linear mesh
#r =np.linspace(0.0001,region,nr)
#log mesh
#a = np.log(2) / (nr - 1)
a = 0.0001
b = radius / (np.e**(a * ( nr - 1)) - 1)
rofi = np.array([b * (np.e**(a * i) - 1) for i in range(nr)])
# make potential
V = makepotential(xx,yy,zz,pot_region,pottype="cubic",potbottom=-1,potshow_f=False)
# surface integral
V_radial = surfaceintegral(x,y,z,rofi,V,method="lebedev_py",potshow_f=False)
vofi = np.array (V_radial) # select method of surface integral
# make basis
node_open = 1
node_close = 2
LMAX = 4
all_basis = []
for lvalsh in range (LMAX):
l_basis = []
# for open channel
val = 1.
slo = 0.
for node in range(node_open):
basis = Basis(nr)
emin = -10.
emax = 100.
basis.make_basis(a,b,emin,emax,lvalsh,node,nr,rofi,slo,vofi,val)
l_basis.append(basis)
# for close channel
val = 0.
slo = -1.
for node in range(node_close):
basis = Basis(nr)
emin = -10.
emax = 100.
basis.make_basis(a,b,emin,emax,lvalsh,node,nr,rofi,slo,vofi,val)
l_basis.append(basis)
all_basis.append(l_basis)
with open ("wavefunc.dat", mode = "w") as fw_w :
fw_w.write("#r, l, node, open or close = ")
for l_basis in all_basis:
for nyu_basis in l_basis:
fw_w.write(str(nyu_basis.l))
fw_w.write(str(nyu_basis.node))
if nyu_basis.open:
fw_w.write("open")
else:
fw_w.write("close")
fw_w.write(" ")
fw_w.write("\n")
for i in range(nr):
fw_w.write("{:>13.8f}".format(rofi[i]))
for l_basis in all_basis:
for nyu_basis in l_basis:
fw_w.write("{:>13.8f}".format(nyu_basis.g[i]))
fw_w.write("\n")
hsmat = np.zeros((LMAX,LMAX,node_open + node_close,node_open + node_close),dtype = np.float64)
lmat = np.zeros((LMAX,LMAX,node_open + node_close,node_open + node_close), dtype = np.float64)
qmat = np.zeros((LMAX,LMAX,node_open + node_close,node_open + node_close), dtype = np.float64)
for l1 in range (LMAX):
for l2 in range (LMAX):
if l1 != l2 :
continue
for n1 in range (node_open + node_close):
for n2 in range (node_open + node_close):
if all_basis[l1][n1].l != l1 or all_basis[l2][n2].l != l2:
print("error: L is differnt")
sys.exit()
hsmat[l1][l2][n1][n2] = integrate(all_basis[l1][n1].g[:nr] * all_basis[l2][n2].g[:nr],rofi,nr) * all_basis[l1][n1].e
lmat[l1][l2][n1][n2] = all_basis[l1][n1].val * all_basis[l2][n2].slo
qmat[l1][l2][n1][n2] = all_basis[l1][n1].val * all_basis[l2][n2].val
print ("\nhsmat")
print (hsmat)
print ("\nlmat")
print (lmat)
print ("\nqmat")
print (qmat)
#make not spherical potential
my_radial_interfunc = interpolate.interp1d(rofi, V_radial)
#V_ang = np.where(np.sqrt(xx * xx + yy * yy + zz * zz) < rofi[-1] , V - my_radial_interfunc(np.sqrt(xx * xx + yy * yy + zz * zz)),0. )
#my_V_ang_inter_func = RegularGridInterpolator((x, y, z), V_ang)
my_V_inter_func = RegularGridInterpolator((x, y, z), V)
#WARING!!!!!!!!!!!!!!!!!!!!!!
"""
Fujikata rewrote ~/.local/lib/python3.6/site-packages/scipy/interpolate/interpolate.py line 690~702
To avoid exit with error "A value in x_new is below the interpolation range."
"""
#!!!!!!!!!!!!!!!!!!!!!!!!!!!
#mayavi.mlab.plot3d(xx,yy,V_ang)
# mlab.contour3d(V_ang,color = (1,1,1),opacity = 0.1)
# obj = mlab.volume_slice(V_ang)
# mlab.show()
#for V_L
fw_umat_vl = open("umat_vl.dat",mode="w")
umat = np.zeros((node_open + node_close,node_open + node_close,LMAX,LMAX,2 * LMAX + 1,2 * LMAX + 1), dtype = np.complex64)
LMAX_k = 9
igridnr = 201
leb_r = np.linspace(0,radius,igridnr)
lebedev_num = lebedev_num_list[-1]
V_L = np.zeros((LMAX_k, 2 * LMAX_k + 1, igridnr), dtype = np.complex64)
leb_x =np.zeros(lebedev_num)
leb_y =np.zeros(lebedev_num)
leb_z =np.zeros(lebedev_num)
leb_w =np.zeros(lebedev_num)
lebedev(lebedev_num,leb_x,leb_y,leb_z,leb_w)
theta = np.arccos(leb_z)
phi = np.where( leb_x **2 + leb_y **2 != 0. , np.where(leb_y >= 0,np.arccos(leb_x / np.sqrt(leb_x **2 + leb_y **2)), np.pi + np.arccos(leb_x / np.sqrt(leb_x **2 + leb_y **2))), 0.)
for i in range(igridnr):
V_leb_r = my_V_inter_func(np.array([leb_x,leb_y,leb_z]).T * leb_r[i]) * leb_w
for k in range(LMAX_k):
for q in range(-k,k+1):
V_L[k][q][i] = 4 * np.pi * np.sum(V_leb_r * sph_harm(q,k,phi,theta).conjugate())
"""
for k in range(LMAX_k):
for q in range(-k,k+1):
print("k = ",k,"q = ", q)
plt.plot(leb_r,V_L[k][q].real,marker=".")
plt.show()
sys.exit()
"""
g_ln = np.zeros((node_open + node_close,LMAX,igridnr),dtype = np.float64)
for n1 in range (node_open + node_close):
for l1 in range (LMAX):
my_radial_g_inter_func = interpolate.interp1d(rofi,all_basis[l1][n1].g[:nr])
g_ln[n1][l1] = my_radial_g_inter_func(leb_r)
C_kq = np.zeros((LMAX,LMAX,2*LMAX+1,2*LMAX+1,LMAX_k,2*LMAX_k+1),dtype=np.float64)
for l1 in range (LMAX):
for l2 in range (LMAX):
for m1 in range(-l1,l1+1):
for m2 in range(-l2,l2+1):
for k in range(1,LMAX_k):
for q in range(-k,k+1):
C_kq[l1][l2][m1][m2][k][q] = (-1) **(-m1) * np.sqrt((2 * l1 + 1) * (2 * l2 +1)) * Wigner3j(l1,0,k,0,l2,0).doit() * Wigner3j(l1,-m1,k,q,l2,m2).doit()
#print(l1,l2,m1,m2,k,q,C_kq[l1][l2][m1][m2][k][q])
count = 0
for l1 in range (LMAX):
for l2 in range (LMAX):
for m1 in range(-l1,l1+1):
for m2 in range(-l2,l2+1):
for n1 in range (node_open + node_close):
for n2 in range (node_open + node_close):
for k in range(1,LMAX_k):
for q in range(-k,k+1):
umat[n1][n2][l1][l2][m1][m2] += simps(g_ln[n1][l1] * V_L[k][q] * g_ln[n2][l2],leb_r) * C_kq[l1][l2][m1][m2][k][q] * np.sqrt((2 * k + 1) / (4 * np.pi))
fw_umat_vl.write("{:>15.8f}".format(count))
fw_umat_vl.write("{:>15.8f}\n".format(umat[n1][n2][l1][l2][m1][m2].real))
count += 1
fw_umat_vl.close()
def test_cg():
cg = CG(1, 2, 3, 4, 5, 6)
wigner3j = Wigner3j(1, 2, 3, 4, 5, 6)
wigner6j = Wigner6j(1, 2, 3, 4, 5, 6)
wigner9j = Wigner9j(1, 2, 3, 4, 5, 6, 7, 8, 9)
assert str(cg) == 'CG(1, 2, 3, 4, 5, 6)'
ascii_str = \
"""\
5,6 \n\
C \n\
1,2,3,4\
"""
ucode_str = \
u("""\
5,6 \n\
C \n\
1,2,3,4\
""")
assert pretty(cg) == ascii_str
assert upretty(cg) == ucode_str
assert latex(cg) == r'C^{5,6}_{1,2,3,4}'
sT(cg, "CG(Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6))")
assert str(wigner3j) == 'Wigner3j(1, 2, 3, 4, 5, 6)'
ascii_str = \
"""\
/1 3 5\\\n\
| |\n\
\\2 4 6/\
"""
ucode_str = \
u("""\
⎛1 3 5⎞\n\
⎜ ⎟\n\
⎝2 4 6⎠\
""")
assert pretty(wigner3j) == ascii_str
assert upretty(wigner3j) == ucode_str
assert latex(wigner3j) == \
r'\left(\begin{array}{ccc} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array}\right)'
sT(wigner3j, "Wigner3j(Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6))")
assert str(wigner6j) == 'Wigner6j(1, 2, 3, 4, 5, 6)'
ascii_str = \
"""\
/1 2 3\\\n\
< >\n\
\\4 5 6/\
"""
ucode_str = \
u("""\
⎧1 2 3⎫\n\
⎨ ⎬\n\
⎩4 5 6⎭\
""")
assert pretty(wigner6j) == ascii_str
assert upretty(wigner6j) == ucode_str
assert latex(wigner6j) == \
r'\left\{\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right\}'
sT(wigner6j, "Wigner6j(Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6))")
assert str(wigner9j) == 'Wigner9j(1, 2, 3, 4, 5, 6, 7, 8, 9)'
ascii_str = \
"""\
/1 2 3\\\n\
| |\n\
<4 5 6>\n\
| |\n\
\\7 8 9/\
"""
ucode_str = \
u("""\
⎧1 2 3⎫\n\
⎪ ⎪\n\
⎨4 5 6⎬\n\
⎪ ⎪\n\
⎩7 8 9⎭\
""")
assert pretty(wigner9j) == ascii_str
assert upretty(wigner9j) == ucode_str
assert latex(wigner9j) == \
r'\left\{\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right\}'
sT(wigner9j, "Wigner9j(Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6), Integer(7), Integer(8), Integer(9))")
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{d}{d x} f{\left (x \right )},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())))")
gg3 = gam3(potargs)
gg4 = gam4(potargs)
zz1 = zeta1(potargs)
zz2 = zeta2(potargs)
zz3 = zeta3(potargs)
zz4 = zeta4(potargs)
nn1 = eta1(potargs)
nn2 = eta2(potargs)
nn3 = eta3(potargs)
kk1 = kappa1(potargs)
kk2 = kappa2(potargs)
kk3 = kappa3(potargs)
# this might overdo it but the sympy expressions
# were handled erroneously before
w3j_m2 = (Wigner3j(1, Lrel - 1, Lrel, 0, 0, 0).doit())**2
w3j_p2 = (Wigner3j(1, Lrel + 1, Lrel, 0, 0, 0).doit())**2
wigm = 0 if (w3j_m2 == 0) else float(w3j_m2.evalf())
wigp = 0 if (w3j_p2 == 0) else float(w3j_p2.evalf())
if pedantic:
print("Lambda = %2.2f A = %d\n" % (Lamb, Ncore))
print("a core = %4.4f\n" % coreosci)
print("LEC 2b = %+6.4e LEC 3b = %+6.4e\n" % (LeC, LeD))
print(' 1 2 3 4')
print("alpha = %+6.4e %+6.4e %+6.4e %+6.4e" % (aa1, aa2,
aa3, aa4))
print("beta = %+6.4e %+6.4e %+6.4e %+6.4e" % (bb1, bb2,
bb3, bb4))
print("gamma = %+6.4e %+6.4e %+6.4e %+6.4e" % (gg1, gg2,
gg3, gg4))
def test_sympy__physics__quantum__cg__Wigner3j():
from sympy.physics.quantum.cg import Wigner3j
assert _test_args(Wigner3j(6, 0, 4, 0, 2, 0))
def test_doit():
assert Wigner3j(1/2, -1/2, 1/2, 1/2, 0, 0).doit() == -sqrt(2)/2
assert Wigner6j(1, 2, 3, 2, 1, 2).doit() == sqrt(21)/105
assert Wigner9j(
2, 1, 1, S(3)/2, S(1)/2, 1, S(1)/2, S(1)/2, 0).doit() == sqrt(2)/12
assert CG(1/2, 1/2, 1/2, -1/2, 1, 0).doit() == sqrt(2)/2