- #1
Septim
- 167
- 6
Greetings,
My task is to prove that the angular momentum operator is hermitian. I started out as follows:
[itex]\vec{L}=\vec{r}\times\vec{p}[/itex]
Where the above quantities are vector operators. Taking the hermitian conjugate yields
[itex]\vec{L''}=\vec{p''}\times\vec{r''}[/itex]
Here I have used double quotes to represent that the hermitian conjugate of the corresponding quantity.
[itex]\vec{L''}=\vec{p}\times\vec{r}[/itex]
Here the fact that the momentum and position are hermitian operators were used. However
[itex]\vec{L''}=\vec{p}\times\vec{r}=-\vec{r}\times\vec{p}=-\vec{L}[[/itex]
There has to be a flaw somewhere but I was not able to catch it, though I was able to prove that the angular momentum operator is hermitian when inspected component by component. I am yet to understand the error in the above derivation. Any help is appreciated.
Thanks in advance
My task is to prove that the angular momentum operator is hermitian. I started out as follows:
[itex]\vec{L}=\vec{r}\times\vec{p}[/itex]
Where the above quantities are vector operators. Taking the hermitian conjugate yields
[itex]\vec{L''}=\vec{p''}\times\vec{r''}[/itex]
Here I have used double quotes to represent that the hermitian conjugate of the corresponding quantity.
[itex]\vec{L''}=\vec{p}\times\vec{r}[/itex]
Here the fact that the momentum and position are hermitian operators were used. However
[itex]\vec{L''}=\vec{p}\times\vec{r}=-\vec{r}\times\vec{p}=-\vec{L}[[/itex]
There has to be a flaw somewhere but I was not able to catch it, though I was able to prove that the angular momentum operator is hermitian when inspected component by component. I am yet to understand the error in the above derivation. Any help is appreciated.
Thanks in advance