Why Does n_0(x) Fail to Satisfy the Spherical Bessel Equation?

In summary, The Spherical Bessel Equation is a mathematical equation used to describe waves in three-dimensional spherical coordinates. It is commonly used in physics to study systems such as atoms, nuclei, and stars. The equation can be solved using different methods and has applications in various fields, including physics and engineering. However, it may not be suitable for all systems and may require numerical approximation in some cases.
  • #1
Logarythmic
281
0
What am I missing when I'm unsuccessful in showing by direct substitution into the spherical Bessel equation

[tex]r^2 \frac{d^2R}{dr^2} + 2r \frac{dR}{dr} + [k^2 r^2 - n(n + 1)] R = 0[/tex]

that

[tex]n_0 (x) = - \frac{1}{x} \sum_{s \geq 0} \frac{(-1)^s}{(2s)!} x^{2s}[/tex]

is a solution?

What's the catch??
 
Last edited:
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  • #2
Ok, if there is no catch, can someone give me at starter here?
 

Related to Why Does n_0(x) Fail to Satisfy the Spherical Bessel Equation?

1. What is the Spherical Bessel Equation?

The Spherical Bessel Equation is a mathematical equation used to describe the behavior of waves that propagate in three-dimensional spherical coordinates. It is a special case of the Bessel differential equation, which is used to describe wave phenomena in cylindrical coordinates.

2. What is the physical significance of the Spherical Bessel Equation?

The Spherical Bessel Equation is commonly used in physics, particularly in electromagnetism and quantum mechanics, to describe the behavior of waves in spherical systems such as atoms, nuclei, and stars. It helps to understand the properties of these systems and predict their behavior under different conditions.

3. How is the Spherical Bessel Equation solved?

The Spherical Bessel Equation can be solved using different methods, such as power series, Frobenius method, or integral representation. The solution depends on the specific boundary conditions and physical system being studied.

4. What are the applications of the Spherical Bessel Equation?

The Spherical Bessel Equation has various applications in physics, including the study of atomic and molecular spectra, diffraction patterns, and the behavior of electromagnetic waves in spherical systems. It is also used in engineering for the analysis of sound and vibration in spherical structures.

5. Are there any limitations to the Spherical Bessel Equation?

The Spherical Bessel Equation may not be suitable for systems with complicated geometries or non-spherical symmetries. In these cases, other equations, such as the Legendre or associated Legendre equations, may be used. Additionally, the solution to the Spherical Bessel Equation may not always have a closed-form expression and may need to be approximated numerically.

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