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Let $T_y(a)$ be a translation operator of a displacement $a$ parallel to the y-axis. In other words,

$$T_y(a)\vec{r}=\vec{r}+a\;\hat{y}$$

If $R_x(\theta)$ is a rotation of $\theta$ around the x-axis, how can I show that

$$R_x(\theta)T_y(a)R_x(-\theta)$$

is a translation along some axis? And how can I determine which axis is it?

[EDIT]

Thanks to mavzolej I was able to determine that the product of those operators generate a translation operator of the type

$$T_{e}(a)\vec{r}= \vec{r} + a\;\hat{e}$$

where $\hat{e}$ is the axis of translation defined by

$$\hat{e} = \cos\theta\;\hat{y} + \sin\theta\;\hat{z}$$

Knowing that, how can I use this result to deduce the commutation relation $[J_x,P_y]=iP_z$ ?

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  • $\begingroup$ Probably, the easiest way would be to use the explicit matrix form. First, learn how the rotation matrices look like in 3d: Section "Basic rotations" here. In order to take translations into account, you will have to slightly generalise the construction, and use 4x4 matrices, see here and here. All together: 1) Write three 4x4 matrices 2) Multiply them 3) Analyse the result. $\endgroup$ – mavzolej May 11 '18 at 6:17
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You can use the definition of J and P as the both are the infinitesimal generators for the both rotation and translation respectively.

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