335 lines
11 KiB
Python
335 lines
11 KiB
Python
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from sympy.core.expr import Expr
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from sympy.core.function import Function, ArgumentIndexError
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from sympy.core.numbers import I, pi
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from sympy.core.singleton import S
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from sympy.core.symbol import Dummy
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from sympy.functions import assoc_legendre
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from sympy.functions.combinatorial.factorials import factorial
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from sympy.functions.elementary.complexes import Abs, conjugate
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import sin, cos, cot
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_x = Dummy("x")
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class Ynm(Function):
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r"""
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Spherical harmonics defined as
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.. math::
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Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}
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\exp(i m \varphi)
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\mathrm{P}_n^m\left(\cos(\theta)\right)
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Explanation
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===========
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``Ynm()`` gives the spherical harmonic function of order $n$ and $m$
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in $\theta$ and $\varphi$, $Y_n^m(\theta, \varphi)$. The four
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parameters are as follows: $n \geq 0$ an integer and $m$ an integer
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such that $-n \leq m \leq n$ holds. The two angles are real-valued
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with $\theta \in [0, \pi]$ and $\varphi \in [0, 2\pi]$.
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Examples
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========
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>>> from sympy import Ynm, Symbol, simplify
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>>> from sympy.abc import n,m
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>>> theta = Symbol("theta")
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>>> phi = Symbol("phi")
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>>> Ynm(n, m, theta, phi)
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Ynm(n, m, theta, phi)
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Several symmetries are known, for the order:
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>>> Ynm(n, -m, theta, phi)
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(-1)**m*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
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As well as for the angles:
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>>> Ynm(n, m, -theta, phi)
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Ynm(n, m, theta, phi)
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>>> Ynm(n, m, theta, -phi)
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exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
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For specific integers $n$ and $m$ we can evaluate the harmonics
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to more useful expressions:
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>>> simplify(Ynm(0, 0, theta, phi).expand(func=True))
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1/(2*sqrt(pi))
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>>> simplify(Ynm(1, -1, theta, phi).expand(func=True))
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sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))
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>>> simplify(Ynm(1, 0, theta, phi).expand(func=True))
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sqrt(3)*cos(theta)/(2*sqrt(pi))
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>>> simplify(Ynm(1, 1, theta, phi).expand(func=True))
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-sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))
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>>> simplify(Ynm(2, -2, theta, phi).expand(func=True))
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sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))
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>>> simplify(Ynm(2, -1, theta, phi).expand(func=True))
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sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))
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>>> simplify(Ynm(2, 0, theta, phi).expand(func=True))
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sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))
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>>> simplify(Ynm(2, 1, theta, phi).expand(func=True))
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-sqrt(30)*exp(I*phi)*sin(2*theta)/(8*sqrt(pi))
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>>> simplify(Ynm(2, 2, theta, phi).expand(func=True))
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sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi))
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We can differentiate the functions with respect
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to both angles:
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>>> from sympy import Ynm, Symbol, diff
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>>> from sympy.abc import n,m
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>>> theta = Symbol("theta")
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>>> phi = Symbol("phi")
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>>> diff(Ynm(n, m, theta, phi), theta)
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m*cot(theta)*Ynm(n, m, theta, phi) + sqrt((-m + n)*(m + n + 1))*exp(-I*phi)*Ynm(n, m + 1, theta, phi)
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>>> diff(Ynm(n, m, theta, phi), phi)
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I*m*Ynm(n, m, theta, phi)
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Further we can compute the complex conjugation:
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>>> from sympy import Ynm, Symbol, conjugate
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>>> from sympy.abc import n,m
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>>> theta = Symbol("theta")
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>>> phi = Symbol("phi")
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>>> conjugate(Ynm(n, m, theta, phi))
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(-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
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To get back the well known expressions in spherical
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coordinates, we use full expansion:
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>>> from sympy import Ynm, Symbol, expand_func
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>>> from sympy.abc import n,m
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>>> theta = Symbol("theta")
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>>> phi = Symbol("phi")
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>>> expand_func(Ynm(n, m, theta, phi))
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sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*exp(I*m*phi)*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))
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See Also
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========
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Ynm_c, Znm
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
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.. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
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.. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
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.. [4] https://dlmf.nist.gov/14.30
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"""
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@classmethod
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def eval(cls, n, m, theta, phi):
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# Handle negative index m and arguments theta, phi
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if m.could_extract_minus_sign():
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m = -m
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return S.NegativeOne**m * exp(-2*I*m*phi) * Ynm(n, m, theta, phi)
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if theta.could_extract_minus_sign():
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theta = -theta
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return Ynm(n, m, theta, phi)
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if phi.could_extract_minus_sign():
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phi = -phi
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return exp(-2*I*m*phi) * Ynm(n, m, theta, phi)
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# TODO Add more simplififcation here
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def _eval_expand_func(self, **hints):
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n, m, theta, phi = self.args
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rv = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
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exp(I*m*phi) * assoc_legendre(n, m, cos(theta)))
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# We can do this because of the range of theta
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return rv.subs(sqrt(-cos(theta)**2 + 1), sin(theta))
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def fdiff(self, argindex=4):
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if argindex == 1:
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# Diff wrt n
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raise ArgumentIndexError(self, argindex)
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elif argindex == 2:
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# Diff wrt m
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raise ArgumentIndexError(self, argindex)
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elif argindex == 3:
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# Diff wrt theta
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n, m, theta, phi = self.args
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return (m * cot(theta) * Ynm(n, m, theta, phi) +
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sqrt((n - m)*(n + m + 1)) * exp(-I*phi) * Ynm(n, m + 1, theta, phi))
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elif argindex == 4:
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# Diff wrt phi
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n, m, theta, phi = self.args
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return I * m * Ynm(n, m, theta, phi)
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else:
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raise ArgumentIndexError(self, argindex)
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def _eval_rewrite_as_polynomial(self, n, m, theta, phi, **kwargs):
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# TODO: Make sure n \in N
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# TODO: Assert |m| <= n ortherwise we should return 0
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return self.expand(func=True)
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def _eval_rewrite_as_sin(self, n, m, theta, phi, **kwargs):
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return self.rewrite(cos)
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def _eval_rewrite_as_cos(self, n, m, theta, phi, **kwargs):
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# This method can be expensive due to extensive use of simplification!
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from sympy.simplify import simplify, trigsimp
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# TODO: Make sure n \in N
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# TODO: Assert |m| <= n ortherwise we should return 0
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term = simplify(self.expand(func=True))
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# We can do this because of the range of theta
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term = term.xreplace({Abs(sin(theta)):sin(theta)})
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return simplify(trigsimp(term))
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def _eval_conjugate(self):
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# TODO: Make sure theta \in R and phi \in R
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n, m, theta, phi = self.args
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return S.NegativeOne**m * self.func(n, -m, theta, phi)
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def as_real_imag(self, deep=True, **hints):
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# TODO: Handle deep and hints
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n, m, theta, phi = self.args
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re = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
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cos(m*phi) * assoc_legendre(n, m, cos(theta)))
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im = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
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sin(m*phi) * assoc_legendre(n, m, cos(theta)))
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return (re, im)
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def _eval_evalf(self, prec):
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# Note: works without this function by just calling
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# mpmath for Legendre polynomials. But using
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# the dedicated function directly is cleaner.
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from mpmath import mp, workprec
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n = self.args[0]._to_mpmath(prec)
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m = self.args[1]._to_mpmath(prec)
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theta = self.args[2]._to_mpmath(prec)
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phi = self.args[3]._to_mpmath(prec)
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with workprec(prec):
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res = mp.spherharm(n, m, theta, phi)
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return Expr._from_mpmath(res, prec)
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def Ynm_c(n, m, theta, phi):
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r"""
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Conjugate spherical harmonics defined as
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.. math::
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\overline{Y_n^m(\theta, \varphi)} := (-1)^m Y_n^{-m}(\theta, \varphi).
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Examples
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========
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>>> from sympy import Ynm_c, Symbol, simplify
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>>> from sympy.abc import n,m
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>>> theta = Symbol("theta")
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>>> phi = Symbol("phi")
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>>> Ynm_c(n, m, theta, phi)
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(-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
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>>> Ynm_c(n, m, -theta, phi)
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(-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
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For specific integers $n$ and $m$ we can evaluate the harmonics
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to more useful expressions:
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>>> simplify(Ynm_c(0, 0, theta, phi).expand(func=True))
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1/(2*sqrt(pi))
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>>> simplify(Ynm_c(1, -1, theta, phi).expand(func=True))
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sqrt(6)*exp(I*(-phi + 2*conjugate(phi)))*sin(theta)/(4*sqrt(pi))
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See Also
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========
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Ynm, Znm
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
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.. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
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.. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
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"""
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return conjugate(Ynm(n, m, theta, phi))
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class Znm(Function):
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r"""
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Real spherical harmonics defined as
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.. math::
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Z_n^m(\theta, \varphi) :=
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\begin{cases}
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\frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &\quad m > 0 \\
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Y_n^m(\theta, \varphi) &\quad m = 0 \\
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\frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &\quad m < 0 \\
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\end{cases}
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which gives in simplified form
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.. math::
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Z_n^m(\theta, \varphi) =
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\begin{cases}
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\frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &\quad m > 0 \\
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Y_n^m(\theta, \varphi) &\quad m = 0 \\
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\frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &\quad m < 0 \\
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\end{cases}
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Examples
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========
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>>> from sympy import Znm, Symbol, simplify
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>>> from sympy.abc import n, m
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>>> theta = Symbol("theta")
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>>> phi = Symbol("phi")
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>>> Znm(n, m, theta, phi)
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Znm(n, m, theta, phi)
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For specific integers n and m we can evaluate the harmonics
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to more useful expressions:
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>>> simplify(Znm(0, 0, theta, phi).expand(func=True))
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1/(2*sqrt(pi))
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>>> simplify(Znm(1, 1, theta, phi).expand(func=True))
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-sqrt(3)*sin(theta)*cos(phi)/(2*sqrt(pi))
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>>> simplify(Znm(2, 1, theta, phi).expand(func=True))
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-sqrt(15)*sin(2*theta)*cos(phi)/(4*sqrt(pi))
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See Also
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========
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Ynm, Ynm_c
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
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.. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
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.. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
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"""
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@classmethod
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def eval(cls, n, m, theta, phi):
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if m.is_positive:
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zz = (Ynm(n, m, theta, phi) + Ynm_c(n, m, theta, phi)) / sqrt(2)
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return zz
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elif m.is_zero:
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return Ynm(n, m, theta, phi)
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elif m.is_negative:
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zz = (Ynm(n, m, theta, phi) - Ynm_c(n, m, theta, phi)) / (sqrt(2)*I)
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return zz
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