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In a question it is given that consider $H-X$ chemical bond and find $H-X$ bond distance for which there is zero probability density of finding the proton. It is easy to solve we have just to calculate the position of nodes of wave function of corresponding harmonic oscillator.

But my question is does the wave function of harmonic oscillator gives the probability density of both electrons and protons of bonded atoms which behaves as harmonic oscillator? As while I was reading the chapter, initially I thought that $\psi$ only gives the probability amplitude of electrons if we extend harmonic oscillator concept to bonded atoms.

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  • $\begingroup$ Hint: The vibration describes the motion of H and X. We haven't mentioned electrons at all. So what does that tell you about what the harmonic oscillator wavefunction is telling you in this case? $\endgroup$
    – Ian Bush
    Apr 26, 2020 at 6:40
  • $\begingroup$ The wave function of two particle oscillating from their equilibrium position(having minimum PE) tells us about the probability amplitude of finding the oscillating particles. But the question which I describe firstly, ask specifically about proton, if it ask for H them it make sense. As two bonded atoms also make a harmonic oscillator system as the get stretched and compressed about their mean position. So does in this case if we use same $\psi$ as that of Harmonic oscillator then it tells specifically about the oscillating atoms or their electrons and protons? $\endgroup$
    – Manu
    Apr 26, 2020 at 6:51
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    $\begingroup$ Complicated problem ultimately - look up Born-Oppenheimer approximation and see if that helps you, strictly there is no such thing as a vibrational wavefunction for a molecule, it is all coupled up together, but it is normally a good approximation to facto the wavefunction into terms that a human can understand, and within that approximation it only describes the motion of the nuclei. Further for H-X for heavy X almost all the motion is the H, so we can further approximate and say X is static, and not the wavefunction is (assuming springs connect the atoms) for SHM of the H atom $\endgroup$
    – Ian Bush
    Apr 26, 2020 at 7:05

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