- Projection Theorem in Spherical Tensor

Consider projecting a vector $\mathbf{v}$ onto a vector $\mathbf{j}$ which is parallel to $\mathbf{v}$, where ‘parallel’ means $\mathbf{v}$ is ‘completely’ represented by $\mathbf{j}$’s basis. The magnitude of the projection of $\mathbf{v}$ onto $\mathbf{j}$ is given by

$$\frac{\mathbf{j}}{\lvert\mathbf{j}\rvert} \cdot \mathbf{v}.$$

Meanwhile, if we denote the projection of $\mathbf{v}$ onto the $q$-th axis as $v_{q}$, then the portion of that projection can also be understood as multiplying the projected magnitude $\mathbf{v}$ has in the direction of $\mathbf{j}$ by the fraction that $\mathbf{j}$ is projected onto the $q$-th axis, namely

$$v_{q} = \left(\frac{\mathbf{j}}{\lvert\mathbf{j}\rvert} \cdot \mathbf{v}\right) \frac{j_q}{\lvert\mathbf{j}\rvert} = \frac{\mathbf{j} \cdot \mathbf{v}}{\lvert\mathbf{j}\rvert^2} \ j_q.$$

, which means, in fact, change of basis.

Similarly, we can write the following relation in the context of spherical tensors

$$\bra{\alpha’, j, m’} \sub{V}{q} \ket{\alpha, j, m}$$

$$= \bra{\alpha’, j, m’} \frac{\mathbf{J}}{\lvert\mathbf{J}\rvert}\cdot \mathbf{V} \ket{\alpha, j, m}\ \frac{\bra{j,m’} \sub{j}{q} \ket{j,m}}{\lvert\mathbf{J}\rvert}$$

$$= \frac{\bra{\alpha’, j, m’} \mathbf{J}\cdot \mathbf{V} \ket{\alpha, j, m}}{\lvert\mathbf{J}\rvert^{2}}\ \bra{j,m’} \sub{j}{q} \ket{j,m}$$

, where

$$\lvert\mathbf{J}\rvert^{2} = \langle j,m \mid \mathbf{J}^{2} \mid j,m \rangle = \hbar^{2} \ j(j+1).$$

Therefore,

$$\langle \alpha’, j, m’ \mid V_{q} \mid \alpha, j, m \rangle$$

$$= \frac{\langle \alpha’, j, m’ \mid \mathbf{J}\cdot \mathbf{V}\mid \alpha, j, m \rangle}{\hbar^{2}\ j(j+1)}\ \langle j,m’ \mid j_{q} \mid j,m \rangle.$$