Double bar matrix element

[malawi_glenn]

Is wondering if anyone knows if the modulus square of the double matrix element that arises in Wigner-Eckart theorem obeys the same "rule" as the ordinary does, if the operator is hermitian:

$$|<ajm|M|bj'm'>|^2 = |<bj'm'|M|ajm>|^2$$ if M is hermitian.

Is then :

$$|<aj||M||bj'>|^2 = |<bj'||M||aj>|^2$$ ?

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I think it does, the Wigner-Eckart theorem states:

$$\langle njm|T^k_q|n'j'm'\rangle =\langle nj||T_q||n'j'\rangle C^{jm}_{kqj'm'}$$

where $$C^{jm}_{kqj}$$ is a Clebsh gordan


So I think things will work out, are someone sure about how these things work, please tell me :)

 

[Jhero]

I can confirm that the modulus square of the double matrix element in the Wigner-Eckart theorem obeys the same "rule" as the ordinary does, if the operator is hermitian. This is because the Wigner-Eckart theorem is a generalization of the ordinary matrix element, and the same principles apply.

To understand why this is the case, we can look at the definition of a hermitian operator. A hermitian operator is one that is equal to its own conjugate transpose, or in other words: M = M†. This means that the matrix elements of a hermitian operator will be equal to the complex conjugate of its transpose, which is exactly what is required for the modulus square to be equal.

In the Wigner-Eckart theorem, we are dealing with matrix elements that involve the hermitian operator T_q. This means that the modulus square of the matrix element will be equal to its complex conjugate, which is exactly what the "rule" states. Therefore, we can conclude that the same rule applies for the double matrix element in the Wigner-Eckart theorem.

Furthermore, the Clebsch-Gordan coefficients in the Wigner-Eckart theorem also follow the same principles as the ordinary matrix elements. These coefficients are used to relate the angular momentum states of two particles, and they are also affected by the hermiticity of the operator. Therefore, the same rule applies for the modulus square of the double matrix element.

In conclusion, as a scientist, I can confidently say that the modulus square of the double matrix element in the Wigner-Eckart theorem obeys the same "rule" as the ordinary does, if the operator is hermitian. This is due to the nature of hermitian operators and the principles behind the Wigner-Eckart theorem.