A Solution to the Schrodinger Equation We can prove that (r)=Ber is a solution,by plugging it into Schrodinger's equation (ignoring angular derivatives): -h2a2平，2aΨ 8mr2ar2+rar厂4e。r eEW EBe or -h2）1 .multiplying through by gives us: 8mat :e2 2 -8m = 4eo「 h2 EBe ar .Now we take the derivatives: aΨ -Be-ar=-aBe-ar 2 102 or or dr2Bear-aBe-wr Ⅲ-7A Solution to the Schrödinger Equation We can prove that Ψ(r) = Be-αr is a solution, by plugging it into Schrödinger’s equation (ignoring angular derivatives): Now we take the derivatives: multiplying through by −h2 8mπ2         −1 gives us: ∂2 Ψ ∂r 2 = ∂2 ∂r 2 Be−αr = α2 Be−αr ∂Ψ ∂r = ∂ ∂r Be−αr = −αBe−αr -h 2 8 m π 2 ∂ 2 Ψ ∂ r 2 + 2 r ∂ Ψ ∂ r      - = E Ψ = EBe e -α r 2 Ψ 4πεo r ∂ 2 Ψ ∂ r 2 + 2 r ∂ Ψ ∂ r         + = -8 m π 2 h 2 EBe 8 m π -α r 2 h 2 e 2 Ψ 4πεo r III-7