Spherical harmonics are mathematical functions that arise in problems with spherical symmetry, particularly in the study of atomic orbitals.

The normalised form of spherical harmonics can be expressed as:

where:

is the polar angle.

is the azimuthal angle.

is the orbital angular momentum quantum number (also known as azimuthal quantum number).

is the magnetic quantum number.

are the associated Legendre polynomials.

is the normalisation constant of the spherical harmonics.

To derive eq400, consider the Schrödinger equation of an electron that is confined to a spherical surface:

where .

Assuming , we can multiply and divide eq401 by and , respectively, to give

The first term on the left of eq402 is independent of . If is varied, only the second and third terms are affected. But the sum of these terms is a constant given by the right-hand side of the equation when is varied. Therefore, we can write

By a similar argument, the first term is a constant when changes, and

The constants and are chosen to be consistent with subsequent derivations. The general solution of eq404 is , where is a constant. To find the specific solution, we substitute in eq404 to give . An eigenfunction must be a non-zero function to have non-trivial solutions. It must also be single-valued to satisfy the Born interpretation. Since , we have and , which after normalisation, becomes . If is single-valued, then .

###### Question

How does show that is single-valued?

###### Answer

A function is considered single-valued if, for every point in its domain, there is a unique value of the function. In mathematical terms, for a given argument , the function should yield a single, well-defined output. Since is the azimuthal angle, which ranges from zero to , the only way to satisfy the single-valued condition is for .

Substituting in yields . As , we have and therefore, . In other words, the normalised solution to eq404 is

To solve eq403, we begin by carrying out the differentiation to give

Like the derivation for the quantum harmonic oscillator, we use a change of variable method, with , to solve eq405.

###### Question

Show that .

###### Answer

Since , we have . Using and the chain rule,

Applying the chain rule again on the first term of RHS of the above equation, or

Substituting , eq406 and eq407 in eq405 yields

Comparing the term with eq50 and eq131a,

Eq408 is known as the associated Legendre differential equation, whose solutions are the associated Legendre polynomials . The product of and gives the spherical harmonics. If , eq408 becomes the Legendre differential equation:

where .

The derivations of the explicit forms of the Legendre polynomials and the associated Legendre polynomials, along with their respective normalisation constants, are described in previous articles. In this article, we offer an alternative approach to demonstrating how Legendre polynomials and the associated Legendre polynomials are related to each other.

Differentiating the Legendre differential equation times using Leibniz’s theorem gives

Assuming that , only the first three terms and the first two terms of the first sum and the second sum, respectively, survive. So, eq409 becomes

where .

If we let , and substitute , and in eq410, we get eq408, where . In other words, we can derive , and hence , if we know .

###### Question

If we have assumed that when we differentiate the Legendre differential equation times to obtain the associated Legendre differential equation, does it mean that in the associated Legendre differential equation are restricted to values greater than or equal to two?

###### Answer

No, the procedure merely demonstrates that the solutions to the associated Legendre differential equation are related to those of the Legendre differential equation by . The allowed values of must be consistent with the solutions to the associated Legendre differential equation. To avoid the trivial solution of , we require that . Thus, can take values of , and while we may choose to differentiate with , the full set of associated Legendre functions exists for .