July 22: Associated Legendre Polynomials (Examples for degree 1 and 2); Space of Spherical Harmonics of Fixed Degree

    Goal: space of spherical harmonics of degree n is an irreducible representation of SO(3).

    We know that the space of spherical harmonics of degree n has dimension 2n+1. The zonal function, say f_n(z), obtained by differentiating \frac{1}{r} by \frac{\partial^n}{\partial z^n}, is a spherical harmonic. We now show that \{ f_n(z), \frac{\partial^k f_n(z)}{\partial \alpha^k}, \frac{\partial^k f_n(z)}{\partial \beta^k} : 0 \leq k \leq n \} (where \alpha, \beta are rotations along the x-axis, y-axis by a small angle, respectively) span the space of spherical harmonics of degree n.

    Let’s take a look at \frac{\partial f_n(z)}{\partial \alpha} (i.e. rotating along the y-axis by a small angle).

    Claim: \frac{\partial}{\partial \alpha} = x \frac{\partial}{\partial z} - z \frac{\partial}{\partial x}.

    Key: differentiating \frac{\partial}{\partial \alpha} is the same as differentiating along the following vector field on the (x, z)-plane:

    Hence, it follows that

    \frac{\partial f_n(z)}{\partial \alpha} = (x\frac{\partial}{\partial z} - z\frac{\partial}{\partial x})f_n(z) = xf'_n(z) (*)

    Hence, (*) = \sqrt{1-z^2}cos(\phi)f'_n(z), where \phi is the angle of rotation along the z-axis.

    Similarly, for the associated Legendre polynomial corresponding to a rotation along the x-axis:

    \frac{\partial f_n(z)}{\partial \beta} = (y\frac{\partial}{\partial z} - z\frac{\partial}{\partial y})f_n(z) = y f'_n(z) = \sqrt{1-z^2}sin(\phi)f'_n(z).

    Notice that differentiating by a small angle of rotation does not change the homogeneity of the polynomial. Since differentiating a harmonic function along a constant vector field still gives us a harmonic function, it follows that \frac{\partial^k f_n(z)}{\partial \alpha^k}, \frac{\partial^k f_n(z)}{\partial \beta^k} are both spherical harmonics of degree n.

    Example: the zonal function of degree 1 is f_1(z) = z.

    The associated functions are:

    z, \sqrt{1-z^2}cos(\phi), \sqrt{1-z^2}sin(\phi).

    Notice that they span the 3-dimensional subspace of spherical harmonics of degree 1.

    For n = 2: the zonal function of degree 2 is f_2(z) = \frac{1}{2}(3z^2-1).

    The associated functions are:

    \frac{\partial f_2(z)}{\partial \alpha} = (x\frac{\partial}{\partial z} - z\frac{\partial}{\partial x})f_2(z) = x \frac{d \frac{1}{2}(3z^2-1)}{dz} = 3zx = 3z\sqrt{1-z^2}cos(\phi)

    \frac{\partial f_2(z)}{\partial \beta} = 3z\sqrt{1-z^2}sin(\phi)

    \frac{\partial^2 f_2(z)}{\partial \alpha^2} = (x\frac{\partial}{\partial z} - z\frac{\partial}{\partial x})(3zx) = 3x^2 - 3z^2 = 3(\sqrt{1-z^2} cos(\phi))^2 - 3z^2 (we obtain an expression that involves cos(2\phi)).

    Similarly, \frac{\partial^2 f_2(z)}{\partial \beta^2} gives us an expression that involves sin(2\phi). The set of functions we generate via these differentiations give us a basis for the space of spherical harmonics of degree 3 (and in general, degree n).

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