Wigner-Eckert Theorem?

by spdf13
Tags: theorem, wignereckert
 P: 265 Wigner-Eckert Theorem? The Wigner-Eckart theorem states that $$\langle J M | T_q^k | J' M'\rangle = (-1)^{J-M} \left(\begin{array}{ccc} J & k & J'\\ -M & q & M'\end{array}\right) \langle J || T^k || J'\rangle$$ It's used for adding two angular momenta: the $M$ and $M'$ dependences are taken out of the matrix elements, simplifying their calculation. The thing in the brackets is a 3-j symbol (i.e. just a number) and the thing on the right is the reduced matrix element. Note that the operator in the reduced matrix element does not depend on $q$ anymore! See example below...
 P: 265 OK, let's look at a very simple irreducible tensor operator (ITO): the $k=1,q=0$ operator: $$T^1_0=J_z[/itex] (the projection of J onto the z-axis) Now, using Wigner-Eckard for two simple spin-1/2 particles ($m=m'=1/2$): [tex]\langle j \frac12 | T_0^1 | j' \frac12\rangle = (-1)^{j-\frac12} \left(\begin{array}{ccc} j & 1 & j'\\ -\frac12 & 0 & \frac12\end{array}\right)\langle j || T^k || j'\rangle$$ In this case, we know from normal quantum mechanics what the matrix element on the left hand side is, so this gives us the reduced matrix element on the right hand side. But for more complicated matrix elements, the procedure is essentially the same. The value of the 3-j symbol can just be found from the definition or from a nice Javasript applet. If all of this made no sense and you're wondering what ITO's are, then I suggest you do some reading. I reccommend the book by Wybourne ("spectroscopic properties of rare earths") or that by Silver ("Irreducible tensor methods"). Don't be discouriged: this stuff is never going to become easy! Cheers, Freek Suyver.