- Vector Operators and Commutation Relations

$$D^{\dagger}(R)V_iD(R) = \sum_j R_{ij}V_j$$

Infinitesimal rotation: #

$$D(R) = e^{-\frac{i}{\hbar}\epsilon \mathbf{J} \cdot \hat{\mathbf{n}}} \approx 1 - \frac{i}{\hbar}\epsilon \mathbf{J} \cdot \hat{\mathbf{n}}$$

Then,

$$\left(1 + \frac{i}{\hbar}\epsilon \mathbf{J} \cdot \hat{\mathbf{n}}\right) V_i \left(1 - \frac{i}{\hbar}\epsilon \mathbf{J} \cdot \hat{\mathbf{n}}\right) = \sum_j R_{ij}V_j$$

Expand the expression: #

$$V_i - \frac{i}{\hbar}\epsilon(\mathbf{J} \cdot \hat{\mathbf{n}})V_i + \frac{i}{\hbar}V_i\epsilon(\mathbf{J} \cdot \hat{\mathbf{n}})$$

$$+ \frac{\epsilon^2}{\hbar^2}(\mathbf{J} \cdot \hat{\mathbf{n}})V_i(\mathbf{J} \cdot \hat{\mathbf{n}}) = \sum_j R_{ij}V_j$$

Neglecting the second-order term of $\epsilon$, we get:

$$V_i + \frac{i}{\hbar}\epsilon[\mathbf{J} \cdot \hat{\mathbf{n}}, V_i] = \sum_j R_{ij}V_j$$

Thus:

$$\epsilon[\mathbf{J} \cdot \hat{\mathbf{n}}, V_i] = \sum_j i\hbar(\delta_{ij} - R_{ij})V_j$$

In case: #

$$\hat{\mathbf{n}} = (0, 0, 1), \quad \mathbf{J} \cdot \hat{\mathbf{n}} = J_3 (= J_z)$$

For small rotations:

$$R = \begin{pmatrix} \cos\epsilon & -\sin\epsilon & 0 \\ \sin\epsilon & \cos\epsilon & 0 \\ 0 & 0 & 1 \end{pmatrix} \approx \begin{pmatrix} 1 & -\epsilon & 0 \\ \epsilon & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Now applying the rotation to the vector:

$$RV = \begin{pmatrix} 1 & -\epsilon & 0 \\ \epsilon & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} V_1 \\ V_2 \\ V_3 \end{pmatrix} = \begin{pmatrix} V_1 - \epsilon V_2 \\ \epsilon V_1 + V_2 \\ V_3 \end{pmatrix}$$

For $i = 1$: #

$$V_1 + \frac{i}{\hbar}\epsilon[J_3, V_1] = V_1 - \epsilon V_2$$

$$[J_3, V_1] = i\hbar V_2$$

For $i = 2$: #

$$V_2 + \frac{i}{\hbar}\epsilon[J_3, V_2] = \epsilon V_1 + V_2$$

$$[J_3, V_2] = -i\hbar V_1$$

For $i = 3$: #

$$[J_3, V_3] = 0$$

Finally, we reach the structure of the commutation relations:

$$[J_i, V_j] = \sum_k i\hbar\epsilon_{ijk}V_k$$