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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 Apr 27 '16 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 Apr 28 '16 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 Apr 28 '16 at 10:49
  • $\begingroup$ @CarloBeenakker Well, by the way I am trying to see where Kendall-Mann numbers’ property and its Gaussian with known error of approximation (MO 164849) could be used. I heard that there is electron diffusion for high energized levels ( probably multiphoton processes in atoms). Can you guess something to apply the property and/or the distribution ( Gaussian for for the finite number N of mixing particles) to physics & an experiment? $\endgroup$ – Mikhail Gaichenkov Apr 28 '16 at 12:25

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