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I'm having a problem proving this operator relation:
[tex]exp(-i\phi\hat{j_{i}})exp(i\theta\hat{j_{k}})exp(i\phi\hat{j_{i}})=exp(i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}})[/tex] (1)
where
[tex][\hat{j_{i}}, \hat{j_{k}}]=i\epsilon_{ikl}\hat{j_{l}}[/tex]. (2)
I can prove this for:
[tex]exp(-i\phi\hat{j_{i}})\hat{j_{k}}exp(i\phi\hat{j_{i}})=cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}}[/tex] (3)
using Baker-Hausdorff lemma.
Now what I do when I'm trying to prove the first expresion, I expand the middle term in Taylor series, and then trying to use this lemma again, but problem arisses with higher powers of [tex]\hat{j_{k}}[/tex].
[tex]exp(-i\phi\hat{j_{i}})(1+i\theta\hat{j_{k}}+\frac{(i\theta\hat{j_{k}})^2}{2!}+\frac{(i\theta\hat{j_{k}})^3}{3!}+...)exp(i\phi\hat{j_{i}})[/tex]
The first term:
[tex]exp(-i\phi\hat{j_{i}})exp(i\phi\hat{j_{i}})=1[/tex]
Second term (what I was able to prove (3)):
[tex]i\theta(exp(-i\phi\hat{j_{i}}))\hat{j_{k}}exp(i\phi\hat{j_{i}})=i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}})[/tex]
And now a problem arisses:
[tex]\frac{(i\theta)^nexp(-i\phi\hat{j_{i}})(\hat{j_{k}})^nexp(i\phi\hat{j_{i}})}{n!}[/tex]
If (1) is true than it should be:
[tex](i\theta)^nexp(-i\phi\hat{j_{i}})(\hat{j_{k}})^nexp(i\phi\hat{j_{i}})/n!=\frac{(i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}}))^n}{n!}[/tex]
becoase
[tex]exp(i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}})=1+(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}})+\frac{(i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}}))^2}{2!}+...[/tex]
but, I can't prove this. Using Baker-Hausdorff lemma for each term becomes too complicated and I get lose in all that mess.
[tex]exp(-i\phi\hat{j_{i}})exp(i\theta\hat{j_{k}})exp(i\phi\hat{j_{i}})=exp(i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}})[/tex] (1)
where
[tex][\hat{j_{i}}, \hat{j_{k}}]=i\epsilon_{ikl}\hat{j_{l}}[/tex]. (2)
I can prove this for:
[tex]exp(-i\phi\hat{j_{i}})\hat{j_{k}}exp(i\phi\hat{j_{i}})=cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}}[/tex] (3)
using Baker-Hausdorff lemma.
Now what I do when I'm trying to prove the first expresion, I expand the middle term in Taylor series, and then trying to use this lemma again, but problem arisses with higher powers of [tex]\hat{j_{k}}[/tex].
[tex]exp(-i\phi\hat{j_{i}})(1+i\theta\hat{j_{k}}+\frac{(i\theta\hat{j_{k}})^2}{2!}+\frac{(i\theta\hat{j_{k}})^3}{3!}+...)exp(i\phi\hat{j_{i}})[/tex]
The first term:
[tex]exp(-i\phi\hat{j_{i}})exp(i\phi\hat{j_{i}})=1[/tex]
Second term (what I was able to prove (3)):
[tex]i\theta(exp(-i\phi\hat{j_{i}}))\hat{j_{k}}exp(i\phi\hat{j_{i}})=i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}})[/tex]
And now a problem arisses:
[tex]\frac{(i\theta)^nexp(-i\phi\hat{j_{i}})(\hat{j_{k}})^nexp(i\phi\hat{j_{i}})}{n!}[/tex]
If (1) is true than it should be:
[tex](i\theta)^nexp(-i\phi\hat{j_{i}})(\hat{j_{k}})^nexp(i\phi\hat{j_{i}})/n!=\frac{(i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}}))^n}{n!}[/tex]
becoase
[tex]exp(i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}})=1+(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}})+\frac{(i\theta(cos(\phi)\hat{j_{k}}+sin(\phi)\hat{j_{l}}))^2}{2!}+...[/tex]
but, I can't prove this. Using Baker-Hausdorff lemma for each term becomes too complicated and I get lose in all that mess.