Question on Spherical Bessel's equation.

[yungman]

Spherical Bessel equation:

$$r^2R''+2rR' + [kr^2-n(n+1)]R = 0 \;\hbox { where }\; k=\lambda^2$$

Boundary condition: $R(a)=0$

Solution :

$$j_n(\lambda_{n,j},r) \;\hbox { where }\; \lambda = \lambda_{n,j}= \frac{\alpha_{n+\frac{1}{2},j}}{a}$$

$$j_n(x)=\sqrt{\frac{\pi}{2x}}J_{n+\frac{1}{2}}(x)$$

I want to find $j_n(\lambda_{n,j}r)$

This is what I have:

$$j_n(\lambda_{n,j}r)=\sqrt{\frac{\pi}{2\lambda_{n,j}r}} \; J_{n+\frac{1}{2}}(\lambda_{n,j}r) = \sqrt{\frac{\pi}{ 2 \frac{\alpha_{(n+\frac{1}{2},j)}}{a}r}} \;\; J_{n+\frac{1}{2}}(\frac{\alpha_{(n+\frac{1}{2},j)}}{a}r)$$

I think I am correct actually the confusion is the order of the Bessel function:

$$p=n+\frac{1}{2} \Rightarrow\; \lambda_p = \lambda_n = \frac{\alpha_{n+\frac{1}{2}}}{a}$$

Can anyone verify this?

 

[jvicens]

I would like to offer my input on this forum post. Based on my knowledge of the spherical Bessel equation, I can say that your solution for j_n(\lambda_{n,j}r) is correct. The order of the Bessel function is indeed p=n+\frac{1}{2}, which means that the parameter \lambda_p is equivalent to \lambda_n. Additionally, the boundary condition R(a)=0 leads to the constraint that \lambda=\lambda_n=\frac{\alpha_{n+\frac{1}{2}}}{a}.

I would also like to mention that the solution you provided, j_n(\lambda_{n,j}r), is the radial solution for the spherical Bessel equation. This means that it represents the behavior of the wave function at different radii r. The full solution for the equation would also include the angular part, which is represented by the spherical harmonics Y_{n}^{m}(\theta, \phi).

Overall, I believe your solution is accurate and well thought out. Keep up the good work!