- Wigner-Eckart theorem
The Wigner-Eckart theorem is a
theorem ofrepresentation theory andquantum mechanics . It states that matrix elements of spherical tensoroperator s on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, while the other is just aClebsch-Gordan coefficient .The Wigner-Eckart Theorem reads
:langle jm|T^k_q|j'm' angle =langle j||T^k||j' angle C^{jm}_{kqj'm'}
where T^k_q is a rank k spherical tensor, jm angle and j'm' angle are eigenkets of total angular momentum J^2 and its z-component J_z, langle j||T^k||j' angle has a value which is independent of m and q, and C^{jm}_{kqj'm'}=langle j'm';kq|jm angle is the Clebsch-Gordan coefficient for adding j' and k to get j.
In effect, the Wigner-Eckart theorem says that operating with a spherical tensor operator of rank k on an angular momentum eigenstate is like adding a state with angular momentum k to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch-Gordan coefficient, which arises when considering adding two angular momenta.
Example
Consider the position expectation value langle njm|x|njm angle. This matrix element is the expectation value of a Cartesian operator in a spherically-symmetric hydrogen-atom-eigenstate basis, which is a nontrivial problem. However, using the Wigner-Eckart theorem simplifies the problem. (In fact, we could get the solution right away using parity, but we'll go a slightly longer way.)
We know that x is one component of vec r, which is a vector. Vectors are rank-1 tensors, so x is some linear combination of T^1_q for q=-1,0,1. In fact, it can be shown that x=frac{T_{-1}^{1}-T^1_1}{sqrt{2. Therefore:langle njm|x|njm angle =frac{1}{sqrt{2langle nj||T^1||nj angle (C^{jm}_{jm11}-C^{jm}_{jm1(-1)})which is zero since both of the Clebsch-Gordan coefficients are zero.
References
*J. J. Sakurai, (1994). "Modern Quantum Mechanics", Addison Wesley, ISBN 0-201-53929-2.
*mathworld|urlname=Wigner-EckartTheorem|title= Wigner-Eckart theorem
* [http://electron6.phys.utk.edu/qm2/modules/m4/wigner.htm Wigner-Eckart theorem]
* [http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/TensorOperators.htm Tensor Operators]
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