SphericalType1BesselFunctionsAndScatteringTheory

The Spherical Bessel Function of the First Kind and Scattering Theory

The spherical Bessel functions of the first kind, denoted as $j_l(x)$, are closely related to scattering theory in quantum mechanics and electromagnetic theory. Here are some key points about their relationship:

1. Radial wave functions: In scattering theory, the spherical Bessel functions of the first kind appear as the radial part of the wave function when solving the Schrödinger equation or Helmholtz equation in spherical coordinates. They describe the radial dependence of the scattered wave.

2. Partial wave expansion: In scattering problems, the incident and scattered waves are often expanded in terms of partial waves using spherical harmonics and spherical Bessel functions. The spherical Bessel functions of the first kind represent the radial part of the incoming and outgoing spherical waves.

3. Phase shifts: The spherical Bessel functions are used to calculate the phase shifts of the partial waves in scattering problems. The phase shifts describe how each partial wave is affected by the scattering potential and are crucial for determining the scattering cross-section and other observables.

4. Scattering amplitude: The scattering amplitude, which determines the probability of scattering in a given direction, is expressed as a sum over partial waves. The coefficients of this sum involve the spherical Bessel functions and the phase shifts.

5. Asymptotic behavior: The asymptotic behavior of the spherical Bessel functions at large distances ($kr >> 1$) is related to the incoming and outgoing spherical waves in the far-field region of the scattering problem. The asymptotic form of $j_l(kr)$ is proportional to $\sin(kr - l\pi/2) / kr$, which represents an outgoing spherical wave.

6. Boundary conditions: The spherical Bessel functions are used to impose boundary conditions on the radial wave functions at the origin and at the boundary of the scattering potential. These boundary conditions ensure the regularity of the wave function and determine the scattering phase shifts.

In summary, the spherical Bessel functions of the first kind play a fundamental role in scattering theory by providing the radial part of the wave functions, enabling the partial wave expansion, and determining the scattering phase shifts and amplitudes. They are essential for describing the behavior of waves in the presence of a scattering potential and for calculating various scattering observables.