Wiegner Eckart theorem, I don't understand it

[Jamalll]

Hello,
In derivation of quadrupole effect (which influences NMR spectrum for nuclei with spin ≥ 1/2), there is one step I do not understand, it is Eckart Wiegner theorem, more specific:

just relavant equations:

Q_αβ=∫(3x_αx_β-δ_αβr²)ρdr

So here is the W-E:

<I,m|Q_αβ|I,m'>=C<I,m|3/2(I_αI_β-I_βI_α)-δ_αβI²|I,m'>

and we expres constant C witm matrix element for m=m'=I, and α=β=z:

eQ=<I,I|Q_zzI,I|>=C<I,I|3I_z²-I²|I,I>=<I,I|I(2I-1)|I,I>



questions:
how can we express Q_αβ with spin operator?
can someone comment on every step what is being done? I would like to understand this intuitively if possible...Great thanks!

 

[Bill_K]

Jamalll, What the Wigner-Eckart Theorem says is that the matrix elements of all quadrupole operators have the same m dependence.

Not saying that Q has any simple relationship to I, it's just that 3/2(I_αI_β + I_βI_α) - δ_αβ is the easiest quadrupole operator to construct and evaluate, and so we use it on the right-hand side to give us something to compare the matrix elements of Q with.

 

[Jamalll]

hey, thanks for reply...


But I still can not make any sense out of it... I know I repeat myself, but
how do you "convert" coordinates in spin operator? What is the significance of "m" and" m' "?

I know it is the state of nucleus with spin "I" and spin projection "m", right?

 

[Jamalll]

Well I am sorry but there is not a single mind which can desolve this?

Baby I am only human!

please...pretty plase