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Schrodinger Equation

What is the r...

Question

What is the ratio of maximum and the middle positions of the radial nodes for an hydrogen atom that has this radial wave equation:
$ψ = k {(\frac{1}{a_{o}})}^{3 / 2} [(x - 1) (x^{2} - 8 x + 12)] e^{- x / 2}$
$x = \frac{2 r}{a_{o}}, a_{o}$ is radius of the first Bohr orbit.

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Solution

At radial node, $ψ = 0$
$ψ = k {(\frac{1}{a_{o}})}^{3 / 2} [(x - 1) (x^{2} - 8 x + 12)] e^{- x / 2} = 0$
From this equation, $(x - 1) (x^{2} - 8 x + 12) = 0$
(i) $(x - 1) = 0$ ; $x = 1$
(ii) $(x^{2} - 8 x + 12) = 0$
$(x - 2) (x - 6) = 0$ ; $x = 2, 6$
So for $ψ = 0; x = 1, 2 o r 6$
Here, $max \to x = 6$
$middle \to x = 2$
Therefore, ratio is 3.

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