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# Klein-Gordon in Spherical Coordinates

Posted 9 years ago
 I need a help. I'm a new user in Mathematica. I need to solve Klein-Gordon in spherical coordinate like this \[Psi]''[r] + 2 /r \[Psi]'[r] + m^2 \[Psi][r] == -E^2 \[Psi][r]  with m is the mass of particle and E is the eigenvalue. The boundary conditions are \[Psi]' == 1, \[Psi] == constant, \[Psi]'[infinity] == 0, \[Psi][infinity] == 0  How is the code that I have to make?
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Posted 9 years ago
 Prof, now I want to solve 2 coupled differential equations with eigenvalue (Ei) like this \[Phi]''[r] + (2/r) \[Phi]'[r] + mb^2 \[Phi][r] + (Ei + g*A[r])^2 \[Phi][r] == 0, A''[r] + (2/r) A'[r] + mv^2 A[r] + 2 g (Ei + g*A[r])^2 (\[Phi][r])^2 == 0 where mb, mv, and g are constants that we assume 1. The boundary conditions of these equations are \[Phi] = 1, \[Phi]' = 0, A = 0, A' = 0. Because the singularity of r, we assume r = 1e-8. and also I want to find the maximum radius (r) when \[Phi][r_max] = 0, \[Phi]'[r_max] = 0, A[r_max] = 0, A'[r_max] = 0. I've tried the code mb = mv = g = 1; b = ParametricNDSolveValue[{\[Phi]''[r] + (2/r) \[Phi]'[r] + mb^2 \[Phi][r] + (Ei + g*A[r])^2 \[Phi][r] == 0, A''[r] + (2/r) A'[r] + mv^2 A[r] + 2 g (Ei + g*A[r])^2 (\[Phi][r])^2 == 0, \[Phi][0.00000001] == 1, \[Phi]'[0.00000001] == 0, A[0.00000001] == 0, A'[0.00000001] == 0}, {\[Phi], A}, {r, 0.00000001, 100}, {Ei}] but it didn't work when I wanted to find Ei by this val = Map[FindRoot[b[Ei], {Ei, #}] &, {1, 2, 3}] and the maximum radius. I want to know where is the problem of my code.
Posted 9 years ago
 Thank you for the answer. I'll try :)
Posted 9 years ago
 This is the radial equation for a free particle. To solve: In:= DSolve[ \[Psi]''[r] + 2/r \[Psi]'[r] + m^2 \[Psi][r] == -E^2 \[Psi][r], \[Psi][r], r] Out= {{\[Psi][r] -> (E^(-Sqrt[-E^2 - m^2] r) C)/r + ( E^(Sqrt[-E^2 - m^2] r) C)/(2 Sqrt[-E^2 - m^2] r)}} C=0 to satisfy the boundary condition at infinity, C can be determined by normalization.