A plane is spanned by the vectors $\vec{a}$ and $\vec{b}$. The angle between the mirror axes $\vec{a^\perp} $ and $\vec{b^\perp}$ is equal to the angle of the two vectors $\vec{a}$ and $\vec{b}$.

Let's say a vector $\vec{u}$ is broken down into its components: $\vec{u} = a \cdot \vec{x} + b \cdot \vec{x^\perp}$

$R_x$ then does the following: $R_x (a \cdot \vec{x} + b \cdot \vec{x^\perp}) = - a \cdot \vec{x} + b \cdot \vec{x^\perp}$

With this knowledge, I now try to deduce how this reflection in the graph comes about:

How comes the green reflection?

Unfortunately, I do not quite understand that. So my ideas are the following: $R_b \vec{c}$ would then negate the $\vec{b}$ component and $R_a (R_b \vec{c})$ would then negate the $\vec{a}$ component of $R_b \vec{c}$. Then my vector would not be as in the picture above left but bottom left.

So my question is, how does this reflection $R_a (R_b \vec{c})$ (picture green) come about? I only have the picture and can not really explain it to me. And if $R_a (R_b \vec{c})$ works as in the picture, how does $-R_a (R_b \vec{v})$ work? And so that I understand correctly, where can you find exactly the mirror axes? Are these orthogonal on $\vec{a}$ and $\vec{b}$?

  • $\begingroup$ It would be helpful if you quoted the original source that your question is about. The diagram uses BRA-KET notation, while you aren’t, so something might have gotten lost in translation. Generally speaking, though, the composition of two reflections is a rotation through twice the angle between the mirror lines, so the illustration is misleading. The resulting vector should indeed be pointing down and left. $\endgroup$ – amd Mar 16 at 20:12
  • $\begingroup$ I agree with your last sentence, if that refers to $R_aR_b|c\rangle$? $\endgroup$ – QuantaMag Mar 18 at 16:55

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