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- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Fowler)/10%3A_Scattering_Theory/10.02%3A_More_Scattering_Theory_-_Partial_WavesWe are considering the solution to Schrödinger’s equation for scattering of an incoming plane wave in the z -direction by a potential localized in a region near the origin. We are, obviously, interest...We are considering the solution to Schrödinger’s equation for scattering of an incoming plane wave in the z -direction by a potential localized in a region near the origin. We are, obviously, interested only in the outgoing spherical waves that originate by scattering from the potential, so we must be careful not to confuse the pre-existing outgoing wave components of the plane wave with the new outgoing waves generated by the potential.
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/08%3A_Central_Potentials/8.02%3A_Infinite_Spherical_Potential_WellThus, the first few spherical Bessel functions take the form \[\begin{aligned} j_0(z) &= \frac{\sin z}{z},\\[0.5ex] j_1(z)&=\frac{\sin z}{z^{\,2}} - \frac{\cos z}{z},\\[0.5ex] y_0(z) &= - \frac{\cos z...Thus, the first few spherical Bessel functions take the form \[\begin{aligned} j_0(z) &= \frac{\sin z}{z},\\[0.5ex] j_1(z)&=\frac{\sin z}{z^{\,2}} - \frac{\cos z}{z},\\[0.5ex] y_0(z) &= - \frac{\cos z}{z},\\[0.5ex] y_1(z) &= - \frac{\cos z}{z^{\,2}} - \frac{\sin z}{z}.\end{aligned}\]
- https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/06%3A_Appendices/6.01%3A_A-_Partial_Wave_AnalysisThe scattering matrix relation can then be re-written as \[\begin{align} c^+_\mu &= c^+_{i,\mu} + c^+_{s,\mu} = \sum_{\mu\nu} S_{\mu\nu} c^-_{i,\nu} \\ \Rightarrow \;\;\; c^+_{s,\ell m} &= 2 \pi \sum_...The scattering matrix relation can then be re-written as \[\begin{align} c^+_\mu &= c^+_{i,\mu} + c^+_{s,\mu} = \sum_{\mu\nu} S_{\mu\nu} c^-_{i,\nu} \\ \Rightarrow \;\;\; c^+_{s,\ell m} &= 2 \pi \sum_{\ell' m'} \Big(S_{\ell m, \ell' m'} - \delta_{\ell \ell'}\delta_{mm'}\Big) e^{i\ell'\pi/2} \, Y_{\ell' m'}^*(\hat{\mathbf{k}}_i)\; \Psi_i.\end{align}\] Using this, the scattered wavefunction can be written as \[\begin{align}\begin{aligned}\psi_s(\mathbf{r}) &= \sum_{\ell m} c^+_{s,\ell m} h_{\ell}…
