Angular momentum ladder operators and state transitions

[JeremyEbert]

What is the significance of the ladder operators eigenvalues as they act on the different magnetic quantum numbers, ml and ms to raise or lower their values?
How do their eigenvalues relate to the actual magnetic transitions from one state to the next?

 

[dextercioby]

Since the 'ladder operators' are not (essentially) self-adjoint, there's no significance of their eigenvalues whatsoever.

 

[JeremyEbert]

Since the 'ladder operators' are not (essentially) self-adjoint, there's no significance of their eigenvalues whatsoever.

Thanks dextercioby. Being very new to the concepts of Hermitian operators, I am obviously having a hard time grasping this explanation but I will continue to research the subject.

My question stems from something I read on OEIS related to NMR spectroscopy. Stanislav Sykora, among other things, maintains a dll for Mnova software. It is used for NMR functionality. On http://oeis.org/A003991, he comments on the intensity of the transition between the states of spin being related to these ladder operators. Is his statement incorrect? If correct, doesn't this give significance to their eigenvalues?

"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]"

 

[dextercioby]

That is correct, but the eigenvalues still mean nothing. The state vectors (kets) are not theirs, but pertain to the spin components (S_z most common).

 

[JeremyEbert]

That is correct, but the eigenvalues still mean nothing. The state vectors (kets) are not theirs, but pertain to the spin components (S_z most common).

Ah, I've heard them described as "in between the state vectors", this makes sense. Is the intensity of the transition between the states referring to the intensity of the magnetic moment? Does this have something to do with the Larmor frequency?

Thanks again for your responses.

 

[JeremyEbert]

Ah, I've heard them described as "in between the state vectors", this makes sense. Is the intensity of the transition between the states referring to the intensity of the magnetic moment? Does this have something to do with the Larmor frequency?

Thanks again for your responses.

I guess my question comes down to;
Can I visualize the ladder operator values as vector rejection values?

http://en.wikipedia.org/wiki/Vector_projection#Vector_rejection_3