- #1
giokara
- 9
- 0
Hi,
I need to let an operator act on a scalar function. The operator is however in a very cryptic form, so I would want to work it out a little bit. I get stuck in the process. The operator is:
[itex]\vec{u}\cdot\left[\vec{L}\times\left(\vec{u}_r\times\vec{L}\right)\right]f[/itex]
Where [itex]\vec{L}[/itex] is the angular momentum operator and [itex]\vec{u}_r[/itex] is the unit vector along the [itex]r[/itex]-direction in a spherical coordinate system. [itex]\vec{u}[/itex] is just a constant vector.
The outer [itex]\vec{L}[/itex] needs to work on both the [itex]\vec{u}_r[/itex] and the inner [itex]\vec{L}[/itex], so I write:
[itex]\vec{u}\cdot\left[\check{\vec{u}_r}\left(\check{\vec{L}}\cdot\vec{L}\right)-\left(\check{\vec{L}} \cdot \check{\vec{u}_r}\right)\vec{L}+\vec{u}_r \left(\check{\vec{L}}\cdot \check{\vec{L}}\right)-\left(\vec{u}_r \cdot \check{\vec{L}}\right)\check{\vec{L}}\right]f[/itex][itex][/itex]
where the upside-down hat denotes the vector on which the [itex]\check{\vec{L}}[/itex] operator acts. In the last three terms the ordering of the operators is correct.
For the first term I do not see how to let [itex]\check{\vec{L}}[/itex] operate on [itex]\check{\vec{u}_r}[/itex] without the second [itex]\vec{L}[/itex] operator acting on it as well. My best guess is to make a tensor term of the sort
[itex]\left(\check{\vec{L}}\check{\vec{u}_r}\right)\cdot\vec{L}[/itex]
but I'm not sure about the ordering of the terms in this expression. I have checked the tensor and it is anti-symmetric, so the ordering will make a difference.
Any suggestions are welcome, thanks in advance!
Giorgos
I need to let an operator act on a scalar function. The operator is however in a very cryptic form, so I would want to work it out a little bit. I get stuck in the process. The operator is:
[itex]\vec{u}\cdot\left[\vec{L}\times\left(\vec{u}_r\times\vec{L}\right)\right]f[/itex]
Where [itex]\vec{L}[/itex] is the angular momentum operator and [itex]\vec{u}_r[/itex] is the unit vector along the [itex]r[/itex]-direction in a spherical coordinate system. [itex]\vec{u}[/itex] is just a constant vector.
The outer [itex]\vec{L}[/itex] needs to work on both the [itex]\vec{u}_r[/itex] and the inner [itex]\vec{L}[/itex], so I write:
[itex]\vec{u}\cdot\left[\check{\vec{u}_r}\left(\check{\vec{L}}\cdot\vec{L}\right)-\left(\check{\vec{L}} \cdot \check{\vec{u}_r}\right)\vec{L}+\vec{u}_r \left(\check{\vec{L}}\cdot \check{\vec{L}}\right)-\left(\vec{u}_r \cdot \check{\vec{L}}\right)\check{\vec{L}}\right]f[/itex][itex][/itex]
where the upside-down hat denotes the vector on which the [itex]\check{\vec{L}}[/itex] operator acts. In the last three terms the ordering of the operators is correct.
For the first term I do not see how to let [itex]\check{\vec{L}}[/itex] operate on [itex]\check{\vec{u}_r}[/itex] without the second [itex]\vec{L}[/itex] operator acting on it as well. My best guess is to make a tensor term of the sort
[itex]\left(\check{\vec{L}}\check{\vec{u}_r}\right)\cdot\vec{L}[/itex]
but I'm not sure about the ordering of the terms in this expression. I have checked the tensor and it is anti-symmetric, so the ordering will make a difference.
Any suggestions are welcome, thanks in advance!
Giorgos