jeremyebert
Can someone explain these "relative intensities" of "transitions between the states of.... spin"

http://oeis.org/A003991

Consider a particle with spin S (a half-integer) and 2S+1 quantum states |m>, m = -S,-S+1,...,S-1,S. Then the matrix element <m+1|S_+|m> = sqrt((S+m+1)(S-m)) of the spin-raising operator is the square-root of the triangular (tabl) element T(r,o) of this sequence in row r = 2S, and at offset o=2(S+m). T(r,o) is also the intensity |<m+1|S_+|m><m|S_-|m+1>| of the transition between the states |m> and |m+1>. For example, the five transitions between the 6 states of a spin S=5/2 particle have relative intensities 5,8,9,8,5. The total intensity of all spin 5/2 transitions (relative to spin 1/2) is 35, which is the tetrahedral number A000292(5). [Stanislav Sykora, May 26 2012]

I think I have it but I would like to see what others think first.
Raphie Frank

35 also makes an appearance as the 3rd row sum of this sequence associated with the Rydberg-Ritz Hydrogen Emission Spectra

Table T(n,k) = k*(4*n+2+k) read by rows.
http://oeis.org/A169603

0 + 11 + 24 = 35

11 is the Co-Totient of 35
24 is the Totient of 35

4pi^2/sqrt 35 ~ 6.67307, which is a nice little approximation of the Gravitational Constant (scaled by a factor of 10^11) and roughly the average of measurements for G dating back to 1969.

- RF
Raphie Frank
QUOTE (jeremyebert+Jul 10 2012, 01:08 AM)
Albers
FUN STUFF. The Klebsch coefficents are bigtime important because they express the probability that dipole transition will take place between energy states. If you look at the integral it is the expression of a dipole moment, namely the integration of PSI*_2 with PSI_1, with the radius vector thrown in between them. I am an electrodynamicist and I am not afraid to say there is a field caused by the beating of the two states which amounts to a dipole antenna, just like a phased array.
Raphie Frank
QUOTE (jeremyebert+Jul 10 2012, 02:54 PM)
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