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It is known that quantum harmonic oscillator is connected to the symmetric group of infinite order which is isomorphic to the permutation group. According to Cayley's theorem any finite group is isomorphic to a subgroup of permutations of elements of this group.

Looking at Kendall-Mann numbers $M(n)$: the maximal number of inversions in a permutation on $n$ letters – we can see Kendall-Mann numbers’ property $M(n)/M(n-1)=n+\frac {1}{2}$ as $n \to \infty$. The same structure is for energy level of the quantum harmonic oscillator.

How can we explain this seemingly coincident based on mathematical approach? What is behind Hamilton operator $H$ and it's $\frac {1}{2}$ expressed as $H=\left(N+\frac {1}{2}\right)\frac {h}{2\pi} \omega$?

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  • $\begingroup$ the answer to your last question (whence the 1/2?) has a simple answer in terms of "zero-point motion", "Maslov index", "WKB turning points" --- I can elaborate, but the Kendall-Mann numbers do not appear. $\endgroup$
    – Carlo Beenakker
    Commented Apr 27, 2016 at 16:24
  • $\begingroup$ @CarloBeenakker Thank you, Does that 'Kendall-Mann numbers do not appear' mean no relation based on math? I try to think about 1/2 in a way which could explain it & "zero-point motion" via the permuatationa & inversions of some mixing particles. Could you have a look at Ben Naim article 'On the Mixing of Diffusing Particles' reffered via the link please? mathoverflow.net/questions/164849/… $\endgroup$
    – Mikhail Gaichenkov
    Commented Apr 28, 2016 at 9:46
  • $\begingroup$ no relationship whatsoever, also notice that the 1/2 offset is not at all special for the harmonic oscillator, any bounded motion with two smooth turning points will give you the same offset. $\endgroup$
    – Carlo Beenakker
    Commented Apr 28, 2016 at 10:49
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    $\begingroup$ physicsforums.com/threads/algebra-of-divergent-integrals.989043 $\endgroup$
    – Anixx
    Commented Jan 2, 2021 at 16:30
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    $\begingroup$ I read your comment at my blog post, Sorry, I can't add any info to that already provided by others here. There are a lot of connections among the Hermite and Laguerre families of polynomials associated with the quantum harmonic oscillator and a variety of combinatorial constructs so that it's plausible there is a connection. $\endgroup$
    – Tom Copeland
    Commented Jul 29, 2022 at 4:47

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