# Wigner-Eckart theorem: Are expectation values of all vector operators parallel?

The Wigner-Eckart theorem tells us that for any tensor operator, $$\mathbf{T}^{(k)}$$ that

\begin{align} \langle jm|T^{(k)}_q|j'm'\rangle = \langle j'm'kq|jm\rangle \langle j||\mathbf{T}^{(k)}||j'\rangle \end{align}

Consider a system in quantum state $$|\psi\rangle = \sum_{j',m'} |j'm'\rangle \langle j'm'|\psi\rangle$$

we then have

\begin{align} \langle T^{(k)}_q\rangle &= \langle \psi|T^{(k)}_q|\psi\rangle = \sum_{j,j',m,m'} = \langle\psi|jm\rangle \langle jm|T^{(k)}_q|j'm'\rangle\langle j'm'|\psi\rangle\\ &= \sum_{j,j',m,m'}\langle \psi|jm\rangle\langle j'm'kq|jm\rangle\langle j||\mathbf{T}^{(k)}||j'\rangle\langle j'm'|\psi\rangle \end{align}

Suppose that the system is in a state of fixed total $$\mathbf{J}$$ so that $$\langle j'm'|\psi\rangle \propto \delta_{j'j_0}$$ so we get

\begin{align} \langle T^{(k)}_q\rangle &= \sum_{m,m'} \langle \psi|j_0m\rangle \langle j_0mkq|j_0m'\rangle \langle j_0||\mathbf{T}^{(k)}||j_0 \rangle \langle j_0m'|\psi\rangle\\ &=\langle j_0||\mathbf{T}^{(k)}||j_0\rangle\sum_{m,m'} \langle \psi|j_0m\rangle\langle j_0mkq|j_0m'\rangle\langle j_0m'|\psi\rangle\\ &=\langle j_0||\mathbf{T}^{(k)}||j_0\rangle A_{kq\psi} \end{align}

Similarly, for a different tensor operator (also of rank $$k$$) we have

$$\langle W^{(k)}_q\rangle = \langle j_0||\mathbf{W}^{(k)}||j_0\rangle A_{kq\psi}$$

We then see that

\begin{align} \langle T^{(k)}_q\rangle &= \frac{\langle j_0||\mathbf{T}^{(k)}||j_0\rangle}{\langle j_0||\mathbf{W}^{(k)}||j_0\rangle} \langle W^{(k)}_q\rangle\\ &= B_{k\mathbf{T}\mathbf{W}j_0} \langle W^{(k)}_q \rangle \end{align}

This tells us more generally that

\begin{align} \langle \mathbf{T}^{(k)}\rangle = B_{k\mathbf{T}\mathbf{W}j_0} \langle \mathbf{W}^{(k)}\rangle \end{align}

This says that the expectation value of any two tensor operators are linearly proportional to eachother. Consider two vector operator $$\mathbf{X} = \mathbf{X}^{(1)}$$ and $$\mathbf{Y} = \mathbf{Y}^{(1)}$$. Then we have that

\begin{align} \langle \mathbf{X}\rangle = B_{1\mathbf{X}\mathbf{Y}j_0}\langle \mathbf{Y}\rangle \end{align}

Telling us that the expcation values of $$\mathbf{X}$$ and $$\mathbf{Y}$$ are linearly proportional. This is surprising to me because classically it is possible to have a system containing vectors pointing in multiple directions. For example, one conclusion of the above analysis is that $$\langle \mathbf{L} \rangle$$ is always proportional to $$\langle \mathbf{S} \rangle$$. I didn't think this was always the case..

Perhaps it is important that I have restricted the system to a state of well defined total $$\mathbf{J}$$. Perhaps when that is done it indicates the system has a special type of rotational symmetry (not generic to all states) which exactly satisfies the above surprising constraints I have demonstrated.

Can anyone please shed some light on the correctness/surprisingness of my conclusions?

Related: Why must the electron's electric dipole moment (EDM) always be aligned with the spin? Also, this question arose from considering this derivation/definition of the Lande-g factor which relies on the fact that $$\langle \mathbf{\mu} \rangle \propto\langle \mathbf{J} \rangle$$.

## 1 Answer

Your conclusion is perfectly correct, but it only applies when the state of the system is known to lie in a single irreducible representation of $$SU(2)$$, i.e. it has a definite total angular momentum $$j$$.

For atomic nuclei, this is often a good approximation, because higher spin states will be substantially higher in energy. At low energies, the nucleus can be regarded as having a definite spin. Then it follows by your argument that all vector operators must be parallel, e.g. $$\langle \mu \rangle \propto \langle \mathbf{J} \rangle$$, even though generally they would be independent. In fact, I'm not sure that very common, quantum mechanics 101 statement makes any sense at all without using the Wigner-Eckart theorem.

The conclusion only sounds strange because most systems do not satisfy the condition. Classical systems have relatively sharply defined angular orientations and hence must occupy many values of $$j$$ by the angular version of the uncertainty principle.