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Show that the hydrogen atomic wavefunction $\psi_{3,1,1}$ is normalized, and that it is orthogonal to $\psi_{3,1,−1}$.

I'm not sure if I'm supposed to consider the radial part. I can show that the spherical part, Y{m=1, l=1}, is normalized and that Y{m = 1, l=1} and Y{m = -1, l=1} are orthogonal... but I'm not sure how to do the radial component if it is supposed to be considered.

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1 Answer 1

To show that the radial part is normalized, you have to integrate its magnitude-squared over the domain, i.e., $r\in[0,\infty)$. But remember that the volume element $\mathrm{d}V\propto r^2\,\mathrm{d}r$, which brings an extra factor of $r^2$, so if you let $x = r/a_0$, up to some multiplicative constants you'll have to evaluate $$\int_0^\infty\left(1-\frac{x}{6}\right)^2x^4e^{-2x/3}\,\mathrm{d}x,$$ the intended answer should be obvious from the radial part of the wavefunction, but deriving it requires some work. I'd recommend using the following commonly useful identity to do so: $$\int_0^\infty x^n e^{-x/b}\,\mathrm{d}x = n!b^{n+1}.$$

As for orthogonality, it's enough to check that the spherical harmonic parts are orthogonal, since the radial part is independent of $\theta,\phi$ and evaluates the same.

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