Question

Determine the raising and lowering operators for the spherical Bessel functions

$R_n{j}_n\left(x\right)=\left(\frac{n}{x}-D\right)j_n\left(x\right)$.

Short answer

$R_n=\left(\frac{n}{x}-D\right)$, is the raising operator of spherical Bessel function anddata-custom-editor="chemistry" $L_n=\left(\frac{n+1}{x}+D\right)$is lowering operator of spherical Bessel function.

Step-by-step solution

 Step 1: Concept of Bessel function

The wave equation is solved in 3D using Bessel functions at a specified (harmonic) frequency. The acoustic pressure at a particular place in 3D space is commonly described by a sum of spherical Bessel functions.

Determine the raising and lowering operators for the spherical Bessel functions 

As given function is$R_n{j}_n\left(x\right)=\left(\frac{n}{x}-D\right)j_n\left(x\right)$,.

Solve the given equation as follows:

$\begin{array}{c}{I}_n{j}_n\left(x\right)=\left(\frac{n+1}{x}+D\right)j_n\left(x\right)\\ R_n{j}_n\left(x\right)=\left(\frac{n}{x}-D\right)j_n\left(x\right)\\ R_n{j}_n\left(x\right)=\left(\frac{n}{x}{j}_n\left(x\right)-j_n^{\text{'}}\left(x\right)\right)\\ R_n{j}_n\left(x\right)=j_{n+1}\left(x\right)\end{array}$

Hence$R_n=\left(\frac{n}{x}-D\right)$, is the raising operator of spherical Bessel function.

$\begin{array}{c}{L}_n{j}_n\left(x\right)=\left(\frac{n}{x}{j}_n\left(x\right)+j_n^{\text{'}}\left(x\right)\right)\\ L_n{j}_n\left(x\right)=j_{n-1}\left(x\right)\end{array}$

Hence,$L_n=\left(\frac{n+1}{x}+D\right)$ is lowering operator of spherical Bessel function.