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Why $1/r$ factor is used?
And in this equation $f_k(\theta,\varphi)$ is scattering amplitude then why plane wave ($e^{ikz}$) amplitude is not used?

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    $\begingroup$ Please use LaTeX to include equations in your post rather than posting images of them. $\endgroup$ Jan 26, 2021 at 5:30
  • $\begingroup$ Because without that factor, the wavefunction won't satisfy scattering Hamiltonian for r trending to infinity $\endgroup$ Jan 26, 2021 at 7:01
  • $\begingroup$ But how? Can u please explain? @aneetkumar $\endgroup$
    – Garry
    Jan 26, 2021 at 7:32

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For large r, the scattering Hamiltonian reduced to the one of a free particle. This wavefunction which is limiting expression of wavefunction for $r\rightarrow \infty$, should satisfy the free particle Hamiltonian. Putting $1/r$ serves the purpose.

Verify it yourself-put this expression in the Hamiltonian of a free particle with momentum $\hbar k$ and see whether the equation is satisfied.

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