"""
Harmonics manages point-group harmonics.
"""
import sympy as sp
from gcoreutils.nsarray import NSArray
from multipie.tag.tag_multipole import TagMultipole
from multipie.harmonics.util.equivalent_operator import equivalent_operator_from_poly
# ==================================================
[docs]
class Harmonics:
"""
point-group harmonics.
Attributes:
tag (TagMultipole): multipole tag.
"""
#
# __v (dict): variable to replace (x,y,z).
# __def (NSArray): definition string (normalized).
# __ex (NSArray): expression string (abc polynomial).
# __umat (NSArray): unitary matrix from Olm to harmonics (2l+1) descending order in m.
#
# ==================================================
def __init__(self, m_tag, def_ex, ex, u):
"""
initialize the class.
Args:
m_tag (TagMultipole or str): multipole tag.
def_ex (str): definition of harmonics in terms of tesseral harmonics.
ex (str): cartesian expression.
u (str): unitary matrix from Olm in descending order in m, [l,l-1,...,-l].
"""
m_tag = TagMultipole(str(m_tag))
self.tag = m_tag
"""multipole tag."""
self.__v = dict(zip(["x", "y", "z"], NSArray.vector3d(self.tag.head).var.values()))
self.__def = NSArray(def_ex)
self.__ex = NSArray(ex)
self.__umat = NSArray(u)
# ==================================================
def __str__(self):
return str(self.tag)
# ==================================================
def __repr__(self):
return repr(self.tag)
# ==================================================
def latex(self):
return self.tag.latex()
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[docs]
def is_axial(self):
"""
axial harmonics ?
Returns:
bool: axial harmonics ?
"""
return self.tag.is_axial
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[docs]
def is_complex(self):
"""
complex harmonics ?
Returns:
bool: complex harmonics ?
"""
return self.tag.is_complex
# ==================================================
[docs]
def definition(self):
"""
definition of harmonics in terms of linear combination of Clm and Slm.
Returns:
NSArray: definition of harmonics.
"""
return self.__def
# ==================================================
[docs]
def expression(self, v=None):
"""
explicit expression of harmonics.
Args:
v (NSArray, optional): vector variable to replace "[x,y,z]".
Returns:
NSArray: polynomial expression of harmonics.
Notes:
- when v is None, default variable is used.
"""
if v is None:
v = self.__v
else:
v = dict(zip(["x", "y", "z"], v))
ex = self.__ex.subs(v)
return ex
# ==================================================
[docs]
def u_matrix(self):
r"""
unitary matrix from (l,m) to the harmonics, :math:`X_{harm} = U^{\dagger}X_{lm}U`.
Returns:
NSArray: unitary matrix from (l,m) to the harmonics with (2l+1) components, descending order in m.
"""
return self.__umat
# ==================================================
[docs]
def to_vector(self):
"""
cartesian vector for rank-1 expression.
Returns:
NSArray: converted cartesian vector.
"""
assert self.tag.rank == 1, "rank is not 1."
ex = self.__ex.subs(
{
"x": sp.Matrix([1, 0, 0]).T,
"y": sp.Matrix([0, 1, 0]).T,
"z": sp.Matrix([0, 0, 1]).T,
}
)
return NSArray(ex[0], "vector")
# ==================================================
[docs]
def equivalent_operator(self, j):
"""
equivalent operator in descending order in Jz.
Args:
j (str): magnitude of angular momentum, J (0, 1/2, 1, ...).
Returns:
NSArray: equivalent-operator matrix.
"""
poly = str(self.expression(v="[x,y,z]"))
return equivalent_operator_from_poly(poly, j)
# ==================================================
[docs]
@classmethod
def create_equivalent_operator(cls, poly, j):
"""
equivalent operator in descending order in Jz.
Args:
poly (str): (x,y,z) polynomial.
j (str): magnitude of angular momentum, J (0, 1/2, 1, ...).
Returns:
NSArray: equivalent-operator matrix.
"""
return equivalent_operator_from_poly(poly, j)
