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Question: Show that the radial wave function for the hydrogen atom are orthogonal with no exceptions.

Attempt: We know that the wave function for the hydrogen atom is orthonormal; that is,

$$\langle\psi_{nml}\mid \psi_{n'm'l'}\rangle=\delta_{nn'}\delta_{mm'}\delta_{ll'}$$

Splitting up $\psi$ into a radial component $R_{nl}$ and a angular component $Y_{lm}$, we have

$$\langle R_{nl}Y_{lm}\mid R_{n'l'}Y_{l'm'}\rangle=\delta_{nn'}\delta_{mm'}\delta_{ll'}$$

We know that the angular component $Y_{lm}$ is orthonormal, and thus

$$\langle Y_{lm}\mid Y_{m'l'}\rangle=\delta_{ll'}\delta_{mm'} $$

I believe that, if we do not have orthogonality in the radial component, we will not have orthogonality in the wave function. For example, if we do not have radial orthogonality, then at $n=n'$, we might have the radial component be zero. Thus, the radial wavefunction must be orthonormal. Is this correct reasoning? i' worried I' missing something.

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