test_spherical_harmonics.py 文件源码

def test_Znm():
    # http://en.wikipedia.org/wiki/Solid_harmonics#List_of_lowest_functions
    th, ph = Symbol("theta", real=True), Symbol("phi", real=True)
    from sympy.abc import n,m

    assert Znm(0, 0, th, ph) == Ynm(0, 0, th, ph)
    assert Znm(1, -1, th, ph) == (-sqrt(2)*I*(Ynm(1, 1, th, ph)
                                  - exp(-2*I*ph)*Ynm(1, 1, th, ph))/2)
    assert Znm(1, 0, th, ph) == Ynm(1, 0, th, ph)
    assert Znm(1, 1, th, ph) == (sqrt(2)*(Ynm(1, 1, th, ph)
                                 + exp(-2*I*ph)*Ynm(1, 1, th, ph))/2)
    assert Znm(0, 0, th, ph).expand(func=True) == 1/(2*sqrt(pi))
    assert Znm(1, -1, th, ph).expand(func=True) == (sqrt(3)*I*sqrt(-cos(th)**2 + 1)*exp(I*ph)/(4*sqrt(pi))
                                                    - sqrt(3)*I*sqrt(-cos(th)**2 + 1)*exp(-I*ph)/(4*sqrt(pi)))
    assert Znm(1, 0, th, ph).expand(func=True) == sqrt(3)*cos(th)/(2*sqrt(pi))
    assert Znm(1, 1, th, ph).expand(func=True) == (-sqrt(3)*sqrt(-cos(th)**2 + 1)*exp(I*ph)/(4*sqrt(pi))
                                                   - sqrt(3)*sqrt(-cos(th)**2 + 1)*exp(-I*ph)/(4*sqrt(pi)))
    assert Znm(2, -1, th, ph).expand(func=True) == (sqrt(15)*I*sqrt(-cos(th)**2 + 1)*exp(I*ph)*cos(th)/(4*sqrt(pi))
                                                    - sqrt(15)*I*sqrt(-cos(th)**2 + 1)*exp(-I*ph)*cos(th)/(4*sqrt(pi)))
    assert Znm(2, 0, th, ph).expand(func=True) == 3*sqrt(5)*cos(th)**2/(4*sqrt(pi)) - sqrt(5)/(4*sqrt(pi))
    assert Znm(2, 1, th, ph).expand(func=True) == (-sqrt(15)*sqrt(-cos(th)**2 + 1)*exp(I*ph)*cos(th)/(4*sqrt(pi))
                                                   - sqrt(15)*sqrt(-cos(th)**2 + 1)*exp(-I*ph)*cos(th)/(4*sqrt(pi)))