# Clebsch-Gordan coefficients and their sign

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• Aleolomorfo
In summary, when combining two isospins with opposite third components, there are two possibilities that give different coefficients due to the sign difference. The two resulting states with ##\Sigma_z=0## have different spin values, with one having a spin of 0 and the other having a spin of 1. This difference is accounted for by taking the square root of all the numbers in the Clebsch-Gordan table, as noted in a footnote.
Aleolomorfo
TL;DR Summary
Different sign in the combination of two ##\textbf{1/2}## isospins with opposite third component
Summary: Different sign in the combination of two ##\textbf{1/2}## isospins with opposite third component

Hello everybody!
I was doing an exercise regarding isospin and I noticed something from the Clebsch-Gordan coefficients that made me think.
For example, if I consider the combination between the two angular momenta ##|\textbf{1/2}, +1/2>## and ##|\textbf{1/2}, -1/2>##, in the Clebsch-Gordan table there are two possibilities. One in which ##I_z=+1/2## is put first and the second in which ##I_z=+1/2## is put after. I have attached the table and I have highlighted what I am saying inside a red circle.
The two combinations gives different coefficients, just for a sign.
My question is: what is the difference between combining ##|\textbf{1/2}, +1/2>## ##|\textbf{1/2}, -1/2>## and ##|\textbf{1/2}, -1/2>## ##|\textbf{1/2}, +1/2>##? Should not they be equal? From what comes the minus sign?

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• 2019-08-09-Note-14-22.pdf
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There are two states with ##\Sigma_z=0##. One is for ##S=0##:
$$|S=0,\Sigma_z=0 \rangle=\frac{1}{\sqrt{2}} (|1/2,1/2 \rangle \otimes |1/2,-1/2 \rangle - |1/2,-1/2 \rangle \otimes |1/2,1/2 \rangle$$
and one for ##S=1##:
$$|S=1,\Sigma_z=0 \rangle=\frac{1}{\sqrt{2}} (|1/2,1/2 \rangle \otimes |1/2,-1/2 \rangle +|1/2,-1/2 \rangle \otimes |1/2,1/2 \rangle.$$
Note the footnote in the Review of Particle Physics table of Clebsch-Gordon coefficients: You have to take the square root of all the number (with the minus-signs outside of the square root)!

## 1. What are Clebsch-Gordan coefficients?

Clebsch-Gordan coefficients are mathematical constants used to calculate the coupling of angular momentum in quantum mechanics. They represent the probability amplitudes for the combination of two angular momenta into a total angular momentum.

## 2. How are Clebsch-Gordan coefficients calculated?

Clebsch-Gordan coefficients can be calculated using the Wigner-Eckart theorem, which relates the matrix elements of a tensor operator to the Clebsch-Gordan coefficients. They can also be found in tables or calculated using computer programs.

## 3. What is the significance of the sign of Clebsch-Gordan coefficients?

The sign of Clebsch-Gordan coefficients indicates the relative phase between the two angular momenta being coupled. It is important in quantum mechanics as it affects the interference patterns of particles and can determine the allowed transitions between energy levels.

## 4. How do Clebsch-Gordan coefficients relate to the addition of angular momenta?

Clebsch-Gordan coefficients are used to calculate the total angular momentum resulting from the addition of two angular momenta. They represent the probability amplitudes for different combinations of the two angular momenta.

## 5. What are some applications of Clebsch-Gordan coefficients?

Clebsch-Gordan coefficients are used in various fields of physics, such as atomic and molecular physics, nuclear physics, and quantum field theory. They are also used in engineering and technology applications, such as in nuclear magnetic resonance spectroscopy and quantum computing.

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