- #1
SplinterCell
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- Homework Statement
- Suppose that the operator ##\hat{V}=\hat{x}\hat{y}\hat{z}## is rotated through an angle ##\theta = \pi / 4## about the ##z##-axis. Find the spherical components of the rotated operator ##\hat{V}^{\prime}##. (Using spherical harmonics with ##\ell = 3## is not allowed).
- Relevant Equations
- The spherical components of the tensor ##T## transform in the following manner: ##T_{q}^{\left(k\right)^{\prime}}=\mathcal{D}\left(R\right)T_{q}^{\left(k\right)}\mathcal{D}^{\dagger}\left(R\right)=\sum_{q^{\prime}=-k}^{k}\mathcal{D}_{q^{\prime}q}^{\left(k\right)}\left(R\right)T_{q^{\prime}}^{\left(k\right)}##
The operator is the ##T_{xyz}## component of the rank 3 tensor ##T=\vec{r}\otimes\vec{r}\otimes\vec{r}## whose Cartesian components are ##T_{ijk}=r_ir_jr_k##. This tensor ##T## also has spherical components ##T_{q}^{(k)}## where ##k=0,1,2,3##, which in principle can be related to their Cartesian counterparts. Each ##r_i## can be written as some linear combination of ##Y_{q}^{(1)}## spherical components. Also, using the properties of the Wigner (small) d-matrix I was able to show that
$$
\mathcal{D}Y_{\pm1}^{\left(1\right)}\mathcal{D}^{\dagger}=e^{\mp i\pi/4}Y_{\pm1}^{\left(1\right)},\;\mathcal{D}Y_{0}^{\left(1\right)}\mathcal{D}^{\dagger}=Y_{0}^{\left(1\right)}
$$
Presumably, if ##\hat{V}=\sum A_{i}Y_{m_{1}}^{\left(1\right)}Y_{m_{2}}^{\left(1\right)}Y_{m_{3}}^{\left(1\right)}## (where ##A_i## are some coefficients) then we can rotate this mess by doing the following:
$$
\hat{V}^{\prime}=\mathcal{D}\hat{V}\mathcal{D}^{\dagger}=\sum A_{i}\mathcal{D}Y_{m_{1}}^{\left(1\right)}\mathcal{D}^{\dagger}\mathcal{D}Y_{m_{2}}^{\left(1\right)}\mathcal{D}^{\dagger}\mathcal{D}Y_{m_{3}}^{\left(1\right)}\mathcal{D}^{\dagger}
$$
If we plug in the previous relations we find that the only thing that changes inside the sum are the coefficients ##A_i## that acquire some phase.
However, I'm not sure how to continue from here. In particular, I don't understand what is meant by the "spherical components of ##\hat{V}^{\prime}##". I mean, ##\hat{V}^{\prime}## is not some vector operator - it's just a (rotated) component of some tensor. Does it even make sense to talk about spherical components?
$$
\mathcal{D}Y_{\pm1}^{\left(1\right)}\mathcal{D}^{\dagger}=e^{\mp i\pi/4}Y_{\pm1}^{\left(1\right)},\;\mathcal{D}Y_{0}^{\left(1\right)}\mathcal{D}^{\dagger}=Y_{0}^{\left(1\right)}
$$
Presumably, if ##\hat{V}=\sum A_{i}Y_{m_{1}}^{\left(1\right)}Y_{m_{2}}^{\left(1\right)}Y_{m_{3}}^{\left(1\right)}## (where ##A_i## are some coefficients) then we can rotate this mess by doing the following:
$$
\hat{V}^{\prime}=\mathcal{D}\hat{V}\mathcal{D}^{\dagger}=\sum A_{i}\mathcal{D}Y_{m_{1}}^{\left(1\right)}\mathcal{D}^{\dagger}\mathcal{D}Y_{m_{2}}^{\left(1\right)}\mathcal{D}^{\dagger}\mathcal{D}Y_{m_{3}}^{\left(1\right)}\mathcal{D}^{\dagger}
$$
If we plug in the previous relations we find that the only thing that changes inside the sum are the coefficients ##A_i## that acquire some phase.
However, I'm not sure how to continue from here. In particular, I don't understand what is meant by the "spherical components of ##\hat{V}^{\prime}##". I mean, ##\hat{V}^{\prime}## is not some vector operator - it's just a (rotated) component of some tensor. Does it even make sense to talk about spherical components?