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The intrinsic magnetic moment of an electron is equal to the magnetic moment that we obtain by effects of orbital motion of a hydrogen electron in the lowest shell (1 Bohr magneton), using classical physics.

Is this merely a coincidence or is there any relation between the quantum mechanical spin moment and the classically obtained orbital moment? How do we reconcile the same value of the two different types of magnetic moments?

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    $\begingroup$ You mean the spin factor $1/2$ cancels against the $g$ factor 2? That's purely numerology, and does not persist at higher order in perturbation theory. $\endgroup$
    – Thomas
    Commented Mar 5, 2018 at 18:01
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    $\begingroup$ @Thomas Yes that is what I mean. Even if it is purely coincidental, it is very intriguing to get the two values as same. Both the values are due to completely different reasons but are the same numerically. $\endgroup$
    – Aniansh
    Commented Mar 5, 2018 at 18:53
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    $\begingroup$ A hydrogen atom has only one electron? Compared to another electron why wouldn’t they be close? $\endgroup$
    – Bill Alsept
    Commented Mar 5, 2018 at 19:27

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The intrinsic magnetic moment of the electron is not equal to its orbital magnetic moment in a hydrogen atom. They agree to lowest order in perturbation theory, but if you calculate them more precisely, you'll see that they are in fact slightly different.

What's more, the fact that to lowest order they agree is nothing but a coincidence. On first principles you can conclude that they are both an integer numbers of $\mu_B$. In this case it just so happens that this integer is the same, but this would not be true if you were considering a bound state with other particles (say, a positron/electron pair or a hydrogen-like atom but with a particle of higher spin).

There is nothing deep going on here, for better or worse.

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    $\begingroup$ Not a coincidence. Dirac predicted the g-factor of the electron. It comes out of his eponymous equation. $\endgroup$
    – Bert Barrois
    Commented Mar 7, 2018 at 16:31
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    $\begingroup$ @BertBarrois sorry but I don't follow the logic. Indeed, Dirac predicted the g-factor of the electron using his equation. How do you conclude from that that the agreement is not a coincidence? $\endgroup$
    – AccidentalFourierTransform
    Commented Mar 7, 2018 at 16:34
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    $\begingroup$ It's not an unexplained adjustable constant like the fine structure constant. It’s not even a mathematical coincidence like ${{\pi }^{2}}\approx 10$. $\endgroup$
    – Bert Barrois
    Commented Mar 7, 2018 at 16:52
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    $\begingroup$ @BertBarrois But note that I never said that it was an unexplained adjustable constant. Quite the contrary, I said that you can prove on first principles that it is an integer times $\mu_B$. I said that the coincidence is that it is the same integer. Again, this needn't be the case; it just so happen to be. $\endgroup$
    – AccidentalFourierTransform
    Commented Mar 7, 2018 at 16:56

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