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Shooting method for wave function

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Nuwan Yapa 2024-11-22 18:21:08 -05:00
parent 0684059359
commit 00a5f66418
1 changed files with 20 additions and 14 deletions
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34
dirac.jl
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@ -25,23 +25,29 @@ end
"Solve the Dirac equation and return the wave function u(r)=[g(r), f(r)] where "Solve the Dirac equation and return the wave function u(r)=[g(r), f(r)] where
divs is the number of mesh divisions so solution would be returned as a 2×(1+divs) matrix, divs is the number of mesh divisions so solution would be returned as a 2×(1+divs) matrix,
refine determines whether to internally enable high-precision mode and optimize the energy beforehand (assuming a bound state), shooting method divides the interval into two partitions at r_max/2, ensuring convergence at both r=0 and r=r_max,
the other parameters are the same from dirac!(...)." the other parameters are the same from dirac!(...)."
function solveWf(κ, p, E, Φ0, W0, B0, A0, r_max, divs; refine=true, normalize=true) function solveWf(κ, p, E, Φ0, W0, B0, A0, r_max, divs; shooting=true, normalize=true)
if refine saveat = r_max / divs
dtype = BigFloat
algo = Feagin12()
f(E_) = boundaryValue(κ, p, E_, Φ0, W0, B0, A0, r_max; dtype=dtype, algo=algo) if shooting
E = find_zero(f, convert(dtype, E)) @assert divs % 2 == 0 "divs must be an even number when shooting=true"
else prob = ODEProblem(dirac!, [0, 1], (r_max, r_max / 2))
dtype = Float64 sol = solve(prob, RK4(), p=(κ, p, E, Φ0, W0, B0, A0), saveat=saveat)
algo = RK() wf_right = reverse(hcat(sol.u...); dims=2)
r_max = r_max / 2 # for the next step
end end
prob = ODEProblem(dirac!, [0, 1], (0, r_max))
sol = solve(prob, RK4(), p=(κ, p, E, Φ0, W0, B0, A0), saveat=saveat)
wf = hcat(sol.u...)
prob = ODEProblem(dirac!, convert.(dtype, [0, 1]), (0, r_max)) if shooting # join two segments
sol = solve(prob, algo, p=(κ, p, E, Φ0, W0, B0, A0), saveat=r_max/divs) rescale_factor = wf[1, end] / wf_right[1, 1]
wf = Float64.(hcat(sol.u...)) rescale_factor ≈ wf[2, end] / wf_right[2, 1] || @warn "Shooting method joint has a discontinuity"
rescale_factor = isfinite(rescale_factor) ? rescale_factor : 1
wf = hcat(wf[:, 1:(end - 1)], wf_right .* rescale_factor)
end
if normalize if normalize
norm = sum(wf .* wf) * r_max / divs # integration by Reimann sum norm = sum(wf .* wf) * r_max / divs # integration by Reimann sum
@ -116,7 +122,7 @@ function calculateNucleonDensity(N, p, Φ0, W0, B0, A0, r_max, divs, E_min=0, E_
ρr2 = zeros(2, divs + 1) # ρ×r² values ρr2 = zeros(2, divs + 1) # ρ×r² values
for (κ, E, occ) in zip(κs, Es, occs) for (κ, E, occ) in zip(κs, Es, occs)
wf = solveWf(κ, p, E, Φ0, W0, B0, A0, r_max, divs; refine=true, normalize=true) wf = solveWf(κ, p, E, Φ0, W0, B0, A0, r_max, divs; shooting=true, normalize=true)
wf2 = wf .* wf wf2 = wf .* wf
ρr2 += (occ / (4 * pi)) * wf2 # 2j+1 factor is accounted in the occupancy number ρr2 += (occ / (4 * pi)) * wf2 # 2j+1 factor is accounted in the occupancy number
end end