Hello! I am reading about the creation and annihilation operators and I don't get how you find the creation operator from the annihilation one. The creation one is(adsbygoogle = window.adsbygoogle || []).push({});

[itex] \hat{a}=\sqrt{\frac{m \omega}{2 \hbar}}\left( \hat{x}+\frac{i \hat{p}}{m \omega}\right) [/itex]

and the annihilation operator is [itex] \hat{a}\dagger =\sqrt{\frac{m \omega}{2 \hbar}}\left( \hat{x}-\frac{i \hat{p}}{m \omega}\right) [/itex]

I don't understand how taking the complex conjugate an transpose leads to the minus sign. I thought that [itex]\hat{x}=x[/itex] and [itex]\hat{p}=i\hbar\frac{\partial}{\partial x}[/itex] so [itex]\hat{x}\dagger=\hat{x}=x[/itex] but

[itex]\hat{p}\dagger=-i\hbar\frac{\partial}{\partial x}=-\hat{p}[/itex] which will cancel with the minus sign from complex conjugating the [itex]i[/itex], so that [itex] \hat{a}=\hat{a}\dagger[/itex].

I'm sure that this mistake is due to my unfamiliarity with the algebra of operators. I would appreciate it if someone could say which of my assumptions is wrong! My hunch is that it has something to do with taking the dagger of [itex]\left( \hat{x}+\frac{i \hat{p}}{m \omega}\right) [/itex] and this not being equal to taking the dagger of the individual operators...

Thanks in advance!

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# Question about creation and annihilation operators?

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