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Why is the binding energy of the electron in atomic hydrogen not the same as the ionization energy of hydrogen? The binding energy is $\approx 13.605874\text{ eV}$ (accounting for fine structure, see also the value in Griffiths'), and the ionization energy is $\approx 13.59844\text{ eV}$ NIST.
The ionization energy for molecular hydrogen is higher at $15.42593(5)\text{ eV}$ NIST.

The difference is about $0.00743\text{ eV}$, which is larger than the Lamb shift and hyperfine structure.

Is the ionization energy lower because it includes excited states of hydrogen? Or is the energy of a free electron lower when other atoms are present - the average radius increases with the principal quantum number, and it cannot increase without bound in the presence of other atoms.

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    $\begingroup$ Just based on different evaluations from different data at different times. Not something to obsess about. $\endgroup$
    – Jon Custer
    Commented Apr 23, 2023 at 1:47
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    $\begingroup$ @JonCuster the difference is larger than the measurement error (presumably), so it surprises me that the NIST listed the value if it is wrong, and that the researcher did not notice. $\endgroup$
    – Yodo
    Commented Apr 24, 2023 at 15:16
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    $\begingroup$ @JonCuster and two upvoters - nope it was the reduced mass correction. This measurement is very precise, and should be in excellent agreement with theoretical predictions. $\endgroup$
    – AXensen
    Commented Apr 24, 2023 at 15:23
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    $\begingroup$ @FlatterMann See my answer. It was missing the reduced mass correction. $\endgroup$
    – AXensen
    Commented Apr 24, 2023 at 15:27
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    $\begingroup$ @Axensen Ooops... then I totally misunderstood the OP in the first place. I thought he was talking about a discrepancy between two different measurements. That makes sense. $\endgroup$ Commented Apr 24, 2023 at 15:29

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You missed the reduced mass correction to the groundstate energy, which is bigger than the fine structure correction (and of course the Lamb shift). In other words, you missed the motion of the proton. Don't despair - you were right to think that the ionization energy should be very precisely measured and in exact agreement with ($-1$ times) the groundstate energy of hydrogen predicted by the schrodinger equation + fine structure + hyperfine structure + lamb shift + proton radius correction + further dirac equation corrections + higher order lamb shift etc.

I derive this result in more detail here (my most popular answer!). IIRC Griffiths doesn't cover this concept in the Hydrogen chapter because it requires discussion of the concept of a two-particle wavefunction, and it doesn't give rise to any interesting degeneracy splitting like fine structure does. Without fine strucure, the groundstate energy of hydrogen is (in SI units): $$ -\frac{\mu e^4}{2(4\pi\epsilon_0)^2\hbar^2} $$ Where $\mu$ is the reduced mass in the center of mass coordinates for the electron and the proton: $$ \mu=\frac{1}{\frac{1}{m_e}+\frac{1}{m_p}}\approx m_e\left(1-\frac{m_e}{m_p}\right) $$ Wolfram alpha gives me the value $13.5982873\text{ eV}$. The error is now well in line with fine structure corrections $O(\alpha^4m_ec^2=0.00015\text{ eV})$.

Finally, note that your mentioning of the molecular hydrogen binding energy is totally irrelevant - it's a completely different (and more complicated) quantum mechanical system, and the interaction between the two hydrogens gives a correction that is $O(1)$ (it cannot be thought of as a small correction to the hydrogen wavefunction).

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  • $\begingroup$ Thanks, I have learnt of this, but forgot. $\endgroup$
    – Yodo
    Commented Apr 26, 2023 at 14:16
  • $\begingroup$ There seem to be two senses of the phrase "Ionization Energy" 1) The energy required to produce a free electron (free the electron from the proton, ie: fully ionized) 2) The energy required to elevate the orbital of electron to the next orbital. Only in sense #1 can "Ionization Energy" == "Binding Energy". Sense #2 is "excitation energy" or, more precisely "minimum excitation energy". Is that 13.5982873 eV? $\endgroup$ Commented Dec 9, 2023 at 18:21
  • $\begingroup$ @JamesBowery Could you clarify more precisely where I used the phrase "ionization energy" to mean the "excitation energy"? I just re-read this answer and I didn't see anywhere where these terms were used incorrectly. As far as I can tell, the "excitation energy" (13eV*(1-1/4)) isn't referred to in any way anywhere in this answer. Perhaps you've missed the key concept that a free electron can have any energy >0, so the ionization energy is exactly equal to the ground state energy times minus one? $\endgroup$
    – AXensen
    Commented Dec 9, 2023 at 18:42
  • $\begingroup$ My apologies as I should have made (most of) that in response to the original question. The portion I really wanted to ask you is better phrased as: "Where can one find the most complete calculation, including all schrodinger equation + fine structure + hyperfine structure + lamb shift + proton radius correction + further dirac equation corrections + higher order lamb shift etc. compared to the best experimental value as well as tolerances for both?" $\endgroup$ Commented Dec 10, 2023 at 19:48
  • $\begingroup$ @JamesBowery I don't fully know. But I can point you in the right direction. The modern theoretical calculations of atomic wavefunctions and energy levels is called "bound state QED." Here is a review paper about it link.springer.com/referenceworkentry/10.1007/…. Then as far as I'm aware the best experimental measurements of hydrogen's energy levels is done by the Hansch group in Munich. www2.mpq.mpg.de/~haensch/htm_neu/research.html . They will be measuring transitions between states, not ionization energy, which is much harder to measure precisely $\endgroup$
    – AXensen
    Commented Dec 11, 2023 at 2:24

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