Wigner-Eckert Theorem?


by spdf13
Tags: theorem, wignereckert
spdf13
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#1
Jan17-04, 02:24 PM
P: 10
Can anyone give me a consice definition of what the Wigner-Eckert Theorem is? I've been reading about it but am finding it difficult to understand. Any help would be appreciated.
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Kurdt
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#2
Jan18-04, 08:58 AM
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Wigner rings a bell with me so I can tell you what his interpretation of Quantum Mechanics was but I'm not sure if that is the Wigner-Eckert theorem. Wigner developed the subjective interpretation of quantum mechanics (as opposed to the Copenhagen interpretation which is the most popular) which explains the collapse of the wavefunctin when something is measured as due to the parallelism of the human consciousness with the rest of the universe. Basically the human conciousness in this interpretation is treated as being separate from the universe, thus when we make a measurement on some quantum system the wavefunction collapses as our mind interprets the information it has received.

I don't know if thats what you're looking for so maybe it will be best if someone else confirms or denies my claims.
spdf13
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#3
Jan18-04, 10:33 AM
P: 10
Thank you but this isn't quite what I had in mind. The wigner-eckart theroem has something to do with the fact that when you want to find the matrix elements of an irreducible tensor operator, you can seperate them into a product of the reduced tensor independent of the quantum number m, and an angular part dependent only on m. This angular part turns out to be the Clebsch-Gordon coefficients. I think you can use this in determining dipole transition probablity amplitudes between quantum states (which is what I'm trying to do).

I'm not completely sure how to implement the procedure, however, and I don't completely understand its importance. If somebody has seen anything like this before can you help me in understanding this theory.

suyver
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#4
Jan19-04, 07:42 AM
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Wigner-Eckert Theorem?


The Wigner-Eckart theorem states that

[tex]\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[/tex]

It's used for adding two angular momenta: the [itex]M[/itex] and [itex]M'[/itex] 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 [itex]q[/itex] anymore!

See example below...
suyver
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#5
Jan19-04, 07:58 AM
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OK, let's look at a very simple irreducible tensor operator (ITO): the [itex]k=1,q=0[/itex] operator:

[tex]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 ([itex]m=m'=1/2[/itex]):

[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[/tex]

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.
spdf13
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#6
Jan19-04, 03:24 PM
P: 10
Thanks for your help everyone. I've been reading a lot of stuff about this in the last couple days and I think I got it. For my problem I am trying to calculate the dipole transition matrix elements for Rb87. After finding all the hyper fine sublevels, I could express my dipole operator as a superpostion of Y1,0, Y1,1 and Y1,-1. From here I inserted this expression into the wigner eckart theorem and come out with a radial part, and an angular part for the different mf levels. This angular part is just the Clebsch gordon coefficients that suyver mentioned. I also think the selection rules will fall out from these integrals. I think I see now how nice the wigner-eckert theorem is. It converts one big mess into two smaller smaller messes, but these two smaller messes are easier to work with.

Again thanks everyone. This is a really nice message board.
vanesch
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#7
Jan19-04, 10:38 PM
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If I may add something: if you have an irreducible tensor operator of rank k, (meaning you have 2k + 1 components), and you want to look at the matrix elements of these components (which are operators) between the kets belonging to a spin-j set and the bras belonging to a spin-j' set, then the W-E theorem states that it is as if you tried to add two spins: spin j and spin k, and you're looking at their spin j' combination, except for a proportionality factor which is a constant if j, j' and k are fixed (the reduced matrix element).

cheers,
Patrick.
Dr. Doak PHD
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#8
Jan23-04, 01:20 AM
P: 1
spdf13,

The Wigner-Eckert Theorem "states" that the matrix element of any spherical tensor between two states of different orientations can occur only (and I stress the word only)under the cloud of conditions precipitating on the strict parameters, limited to the event contained in the whereabouts when the twin paired reduced matrix element's magnetic movement is parallel to, and completely in sync with that magnetic movement of the G-factored figment particle.

Dr. Doak PHD @ The Celiquintial Research Lab


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