Wigner–Eckart theorem

The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators 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, and the other a Clebsch-Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.^[1]

The Wigner-Eckart theorem reads:

$\langle jm|T^k_q|j'm'\rangle =\langle j||T^k||j'\rangle C^{jm}_{kqj'm'}$

where $T^k_q$ is a rank $\!k$ spherical tensor, $|jm\rangle$ and $|j'm'\rangle$ are eigenkets of total angular momentum $\!J^2$ and its z-component $\!J_z$ , $\langle j||T^k||j'\rangle$ has a value which is independent of $\!m$ and $\!q$ , and $C^{jm}_{kqj'm'}=\langle j'm';kq|jm \rangle$ 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 $\langle njm|x|njm\rangle$ . 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 $x=\frac{T_{-1}^{1}-T^1_1}{\sqrt{2}}$ , where we defined the spherical tensors^[2] $T^1_{0}=z$ and $T^1_{\pm1}=\mp (x \pm i y)/{\sqrt{2}}$ (the pre-factors have to be chosen according to the definition^[3] of a spherical tensor of rank $k$ . Hence, the $T^1_{q}$ are only proportional to the ladder operators). Therefore

$\langle njm|x|n'j'm'\rangle = \langle njm|\frac{T_{-1}^{1}-T^1_1}{\sqrt{2}}|n'j'm'\rangle = \frac{1}{\sqrt{2}}\langle nj||T^1||n'j'\rangle (C^{jm}_{1(-1)j'm'}-C^{jm}_{11j'm'})$

The above expression gives us the matrix element for $x$ in the $|njm\rangle$ basis. To find the expectation value, we set $n'=n$ , $j'=j$ , and $m'=m$ . The selection rule for $m'$ and $m$ is $m\pm1=m'$ for the $T_{\mp1}^{(1)}$ spherical tensors. As we have $m'=m$ , this makes the Clebsch-Gordan Coefficients zero, leading to the expectation value to be equal to zero.

References

^ Eckart Biography– The National Academies Press
^ J. J. Sakurai: "Modern quantum mechanics" (Massachusetts, 1994, Addison-Wesley)
^ J. J. Sakurai: "Modern quantum mechanics" (Massachusetts, 1994, Addison-Wesley)

External links

J. J. Sakurai, (1994). "Modern Quantum Mechanics", Addison Wesley, ISBN 0-201-53929-2.
Weisstein, Eric W., "Wigner–Eckart theorem" from MathWorld.
Wigner–Eckart theorem
Tensor Operators