Derive the equation for the Laplacian operator in spherical coordinates using the gradient operator in spherical coordinates
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SOLVED: The Laplacian in spherical coordinates is ∇^2 = 1/r^2 ∂/∂r (r^2 ∂/∂r) 1/(r^2 sinθ) ∂/∂θ (sinθ ∂/∂θ) 1/(r^2 sin^2θ) ∂^2/∂φ^2, where ∇^2 is a differential operator acting on angular coordinates θ and φ. Consider the Helmholtz equation in three dimensions: ∇^2ψ k^2ψ = 0. This equation appears often in wave mechanics. Use separation of variables via the ansatz ψ(r, θ, φ) = R(r)Θ(θ)Φ(φ), and assume that Θ satisfies LΘ(θ) = -λ(λ 1)Θ(θ). S
SOLVED: The Laplacian in spherical coordinates is ∇^2 = 1/r^2 ∂/∂…
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