Recursion relation

To solve the radial part of the Schrödinger equation for a spherically symmetric potential, we often encounter the need to solve the angular part separately. This typically involves solving for the angular momentum eigenfunctions, which are characterized by the quantum numbers l and m_l​.

One common technique for solving the angular part involves separation of variables and leads to the spherical harmonics.

When considering the addition of angular momentum, say for two particles or two angular momenta, one often uses Clebsch-Gordan coefficients. These coefficients express the total angular momentum state ∣in terms of the individual angular momentum states. They satisfy certain orthogonality and completeness relations.

The recursion relations often arise when we calculate these Clebsch-Gordan coefficients. They provide a way to calculate the coefficients for higher angular momentum states using those for lower angular momentum states. The specific form of the recursion relation depends on the conventions used for the Clebsch-Gordan coefficients.

We can write

From 1 and 2

Here J² = J_1 + J_2

We consider the case of j_2 = 1/2

For j_2 = 1/2 m_2 = 1/2 and -1/2

First we consider j_2 = 1/2 and m_2 = 1/2

On LHS of equation 3 , m_1 + m_2 = 1/2

m_1 = m-1-m_2 = m-1-1/2 = m-3/2

Putting m_1 = m-3/2 and j_2 = 1/2 in equation 3 we obtain

Recursion relation are indispensable tools in quantum mechanics for understanding the addition of angular momenta and determining the possible states of composite quantum systems.

This note is a part of the Physics Repository.