Wigner-Eckart theorem

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The Wigner-Eckart theorem is a theorem of representation theory and quantum mechanics allowing operators to be transformed from one basis to another. These transformations involve the use of Clebsch-Gordan coefficients.

In Quantum Mechanics, the equation associated with the Wigner-Eckart Theorem is:

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

where T^k_q is a rank q spherical tensor, |njm\rangle and |n'j'm'\rangle are eigenkets of J2 and Jz, \langle nj||T_q||n'j'\rangle has a value which is independent of m and C^{jm}_{kqj'm'} is a Clebsch-Gordan coefficient.

[edit] Example

Consider the position expectation value \langle njm|x|njm\rangle. Now we know that x goes like T^{-1}_{1}-T^1_1. Therefore

\langle njm|x|njm\rangle =\frac{1}{\sqrt{2}}\langle nj||T_1||n'j'\rangle (C^{jm}_{jm11}-C^{jm}_{jm1-1})

which is zero since both of the Clebsch-Gordan Coefficients are zero.

[edit] References

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