# Sakurai proof Wigner-Eckart theorem

• detste
In summary, the Wigner-Eckart theorem can be found on page 240 in Sakurai's book "Modern Quantum Mechanics" (Revised edition). The proof of the theorem shows that the recursion relation for the Tq(k) matrix elements is the same as the recursion relation for the Clebsch-Gordon coefficients. This means that the ratios between the Tq(k) matrix elements are the same as those between the Clebsch-Gordon coefficients. However, the absolute magnitude of the Tq(k) matrix elements can vary and is not determined by the recursion relation. Therefore, the only difference between the Tq(k) matrix elements and the Clebsch-Gordon coefficients is an overall scale factor, which does not depend on
detste
On page 240 in Sakurai's book "Modern Quantum Mechanics" (Revised edition) you can find a proof of the Wigner-Eckart theorem. I don't understand how you can deduce equation (3.10.36) from equation (3.10.35). I also don't understand why this proportionality factor is independent of m, q and m'. Can someone clarify this?

Well first of all, 3.10.36 follows from 3.10.35 because the recursion relation for the Tq(k) matrix elements is the same as the recursion relation for the Clebsch-Gordon coefficients.

The recursion relation for the CG coefficients gives us the ratio between any two CG coefficients. But there is nothing which dictates their absolute magnitudes (aside from an arbitrary convention). If the Tq(k) matrix elements are known to follow the same recursion relation, then we know the ratios between all of them, too, and these ratios are exactly the same as those between the CG coefficients. However, the absolute magnitude of these matrix elements is not dictated by the recursion relation, and the absolute magnitude need not be equal to that of the CG coefficient. Thus the only difference that can exist between the Tq(k) matrix elements and the CG coefficients is an overall scale factor (by overall I mean the same scale factor relates all Tq(k) matrix elements with their associated CG coefficient.)

Now since the CG coefficient recursion relation dictates all of the CG coefficients (not just for a given m, q, or m'), then this overall scale factor must not depend on m, q, or m'.

## 1. What is the Wigner-Eckart theorem?

The Wigner-Eckart theorem is a mathematical theorem in quantum mechanics that relates the matrix elements of operators between different states of a quantum system.

## 2. How does the Wigner-Eckart theorem relate to Sakurai proof?

The Sakurai proof is a specific mathematical proof of the Wigner-Eckart theorem, developed by physicist Jun John Sakurai. It provides a more direct and elegant approach to proving the theorem compared to the original proof by Eugene Wigner and Friedrich Eckart.

## 3. What is the significance of the Wigner-Eckart theorem?

The Wigner-Eckart theorem is an important tool in quantum mechanics, as it allows for the simplification and prediction of matrix elements, making calculations easier and more efficient. It also has applications in other fields such as nuclear physics and molecular spectroscopy.

## 4. How is the Wigner-Eckart theorem used in practical applications?

The Wigner-Eckart theorem is used in various practical applications, such as predicting the strengths of atomic and molecular transitions, calculating magnetic moments in nuclear physics, and determining the selection rules for allowed transitions in spectroscopy.

## 5. Are there any limitations to the Wigner-Eckart theorem?

While the Wigner-Eckart theorem is a useful tool, it does have limitations. It only applies to systems that have rotational symmetry, and it does not take into account the effects of spin-orbit coupling. Additionally, it may not be applicable to systems with more than one particle or with strong interactions.

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