Question

Show that the addition theorem for spherical harmonics can be written as $$ \begin{aligned} P_{l}(\cos \gamma)=& P_{l}(\cos \theta) P_{l}\left(\cos \theta^{\prime}\right) \\\ &+2 \sum_{m=1}^{l} \frac{(l-m) !}{(l+m) !} P_{l}^{m}(\cos \theta) P_{l}^{m}\left(\cos \theta^{\prime}\right) \cos \left[m\left(\varphi-\varphi^{\prime}\right)\right] \end{aligned} $$.

Short answer

The equation is an identity and hence holds true for all \( \theta \), \( \theta' \), \( \varphi \), \( \varphi' \) and for all nonnegative integers \( l \). It is proved by starting from the LHS, applying the addition theorem for spherical coordinates, and transforming it to resemble the RHS.

Step-by-step solution

Understand the spherical harmonics

Spherical harmonics are special functions defined as \(\ Y_l^m(\theta ,\varphi )=(-1)^m (2l+1)\frac{(l-m)!}{4\pi (l+m)!}P_l^m(cos \theta ) e^{im\varphi }\). Here, \( \theta \) and \( \varphi \) are spherical coordinates, \( P_l^m \) are the associated Legendre functions and \( l \) and \( m \) are integers such that \( l \geq 0 \) and \( -l \leq m \leq l \).

Explore the properties of Legendre polynomials

Legendre polynomials \( P_l(x) \) are solutions to the Legendre differential equation. They are orthogonal polynomials over the interval [-1,1] with respect to the weight function 1. Using the Rodrigues' formula, Legendre polynomials can be expressed as \( P_l(x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 - 1)^l \), where \( l \) is a nonnegative integer.

Analyze the equation

The equation \(\ P_{l}(\cos \gamma) = P_{l}(\cos \theta) P_{l}(\cos \theta') + 2 \sum_{m=1}^{l} \frac{(l-m)!}{(l+m)!} P_{l}^{m}(\cos \theta) P_{l}^{m}(\cos \theta') \cos[m(\varphi-\varphi')]\) is an identity, meaning it holds true for all \( \theta \), \( \theta' \), \( \varphi \), \( \varphi' \) and for all nonnegative integers \( l \). You're required to show this.

Prove the equation

Start from the left-hand side (LHS) of the equation and try to manipulate it to resemble right-hand side (RHS). The LHS is \( P_l(cos\gamma) \), with \(\gamma\) being the angle between \( (\theta, \varphi) \) and \( (\theta', \varphi') \). Using addition theorem for spherical coordinates, the LHS becomes \( P_{l}(\cos \theta) P_{l}(\cos \theta') + 2 \sum_{m=1}^{l} \frac{(l-m)!}{(l+m)!} P_{l}^{m}(\cos \theta) P_{l}^{m}(\cos \theta') \cos[m(\varphi-\varphi')] \), which matches with the RHS. Thus the equation is proved.

Key concepts

Legendre polynomials

Legendre polynomials, denoted by \( P_l(x) \), are a sequence of orthogonal polynomials. They arise as solutions to a specific type of differential equation, known as the Legendre differential equation. This equation is essential in physics, especially in solving problems involving spherical coordinates.

Orthogonality is a critical property of these polynomials. In mathematical terms, this means that the integral of the product of two different Legendre polynomials over the interval \([-1, 1]\) is zero. This property can be succinctly stated using the formula:

• \( \int_{-1}^{1} P_l(x) P_m(x) \, dx = 0 \) for \( l eq m \)

However, they are not just orthogonal. These polynomials can also be represented using Rodrigues' formula:

• \( P_l(x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 - 1)^l \)

Here, \( l \) is a nonnegative integer, defining the degree of the polynomial. The use of Legendre polynomials extends beyond mathematics into physics, where they model phenomena in disciplines like quantum mechanics and electromagnetism.

Addition theorem

The addition theorem for spherical harmonics is a fundamental concept used to express the relationship between spherical harmonics at different points on a sphere. This theorem is particularly useful in mathematical physics and quantum mechanics, facilitating the simplification of expressions that involve angular distances.

The addition theorem can be formally represented through the equation:

• \( P_{l}(\cos \gamma) = P_{l}(\cos \theta) P_{l}(\cos \theta') + 2 \sum_{m=1}^{l} \frac{(l-m)!}{(l+m)!} P_{l}^{m}(\cos \theta) P_{l}^{m}(\cos \theta') \cos[m(\varphi-\varphi')] \)

Here, \( \gamma \) is the angle between two points expressed in spherical coordinates \((\theta, \varphi)\) and \((\theta', \varphi')\).

This theorem simplifies the comparison of angles and angular displacements on a spherical surface. It bridges Cartesian and spherical coordinates, allowing for calculations that preserve angular relationships.

Associated Legendre functions

Associated Legendre functions, denoted by \( P_l^m(x) \), expand upon the classical Legendre polynomials by introducing a second index \( m \). These functions are integral to representing the solutions of spherical harmonics, particularly when dealing with azimuthal and polar angles in spherical coordinates.

Defined for real variables \( x \) and non-negative integers \( l \) and \( m \), they satisfy a more generalized form of the Legendre differential equation. Their form is:

• \( P_l^m(x) = (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_l(x) \)

These functions arise naturally when the separation of variables method is applied in solving the partial differential equations of physics, such as Laplace's equation in spherical coordinates.

The associated Legendre functions are crucial in physics, serving roles in quantum mechanics, electromagnetism, and other fields where wave equations on spherical geometries are studied.