Parabolic Designs
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SOLVED:The parabolic coordinates (ξ, η, ϕ) of a point in three-dimensional space are defined by the relations x =√(ξη) cosϕ, ξ=r z y =√(ξη) sinϕ, η=r-z z =(1)/(2)(ξ-η), ϕ=tan^-1(y / x) r =(1)/(2)(ξ η) and the Laplacian operator is given in these coordinates by ∇^2=(4)/(ξ η)[(∂)/(∂ξ)(ξ(∂)/(∂ξ)) (∂)/(∂η)(η(∂)/(∂η))] (1)/(ξη) (∂^2)/(∂ϕ^2) (a) Prove that the Schrödinger equation (7.94) with a Coulomb potential is separable in these coordinates. (Hint: W
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