The Spatial part of the Time-Independent Schrödinger Equation (TISE) is

\left(- \frac {\hbar^2}{2m} \nabla^2 +V(r) \right) \psi(r) = E \psi(r)

by setting

k^2 = 2m E / \hbar^2   and    U(r) = 2 m V(r) /\hbar^2

the equation becomes a wave equation with a source, or scattering equation.

(\nabla^2 +k^2) \psi(r) = U(r) \psi(r)

for solving it, we have to find the Green function such that

(\nabla^2 + k^2 ) G(r,r') = \delta(x-x')

The solution is easy ( i will post it later, you can think the Green function is the inverse of the Operator)

G(r,r') = - \frac{1}{4\pi} \frac {Exp( \pm i k \cdot (r-r') )} { |r - r'| }

the particular solution is

\psi_p(r) = \int {G(r,r') U(r') \psi(r') dr'}

plus the homogeneous solution

\psi(r) = \phi(r) +\int {G(r,r') U(r') \psi(r') dr'}

but it is odd that the solution contain itself!

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