Let's first write (stationary) Schrodinger equation in cartesian coordinates :
−2mℏ2(∂x2∂2+∂y2∂2+∂z2∂2)ψ+Vψ=Eψ
−2mℏ2Δψ+(V−E)ψ=0 , where Δ is Laplace operator
Δψ+ℏ22m(E−V)ψ=0
Now it is just enough to use the expression of Δ in spherical coordinates :
r21∂r∂(r2∂r∂ψ)+r2sinθ1∂θ∂(sinθ∂θ∂ψ)+r2sin2θ1∂ϕ2∂2ψ+ℏ22mψ=0
The expression of Δ in spherical coordinates can be found, for example, in Wikipedia : Laplace operator - Wikipedia . We can also, of course, calculate it directly :
∂x∂=∂x∂r∂r∂+∂x∂θ∂θ∂+∂x∂ϕ∂ϕ∂ (chain rule)
∂x∂=sinθcosϕ∂r∂−rcosθcosϕ∂θ∂−rsinθsinϕ∂ϕ∂ (using the expressions of spherical coordinates in cartesian coordinates)
The same calculation for y,z gives :
∂y∂=sinθsinϕ∂r∂−rcosθsinϕ∂θ∂+rsinθcosϕ∂ϕ∂
∂z∂=cosθ∂r∂−rsinθ∂θ∂
And now we find by direct calculation (as ∂x2∂2=(∂x∂)2 and same for other coordinates) :
Δ=r21∂r∂(r2∂r∂)+r2sinθ1∂θ∂(sinθ∂θ∂)+r2sin2θ1∂ϕ2∂2
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