Different components of angular momentum do not commute

[Prometheus]

I'm having a little trouble with some commutational relations between different components of the angular momentum operators. I'm OK up to:

 

[math][\hat{L_x},\hat{L_y}]=\hat{Y}\hat{P_z}\hat{Z}\hat{P_x} - \hat{Z}\hat{P_x}\hat{Y}\hat{P_z} + \hat{Z}\hat{P_y}\hat{X}\hat{P_z} - \hat{X}\hat{P_z}\hat{Z}\hat{P_y}[/math]

 

and the apparently this is equal to

 

[math]\hat{Y}\hat{P_x}(\hat{P_z}\hat{Z} - \hat{Z}\hat{P_z}) + \hat{X}\hat{P_y}(\hat{Z}\hat{P_z} - \hat{P_z}\hat{Z})[/math]

 

I don't understand this 'factoring out' of operators. I thought it was a typo in my notes and the answer is:

 

[math]\hat{Y}\hat{P_z}(\hat{Z}\hat{P_z} - \hat{Z}\hat{P_z}) + \hat{X}\hat{P_y}(\hat{Z}\hat{P_z} - \hat{Z}\hat{P_z})[/math]

 

but then the rest of the derivation doesn't work.

 

I think i'm missing some property of commutators, but can't see what it is.

 

Help appreciated.

[Mordred]

This may help.

https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www2.ph.ed.ac.uk/~ldeldebb/docs/QM/lect8.pdf&ved=0ahUKEwiw-8a6zKnRAhVcVWMKHQMzC50QFggvMAQ&usg=AFQjCNHaBacLpJQ-41rXS0L58bK_m3Y3BA&sig2=p0G0kk7RP3Dyd8tlwjGq7w

http://farside.ph.utexas.edu/teaching/qmech/Quantum/node71.html

Actually this article may be better in details.

https://www.google.ca/url?sa=t&source=web&rct=j&url=https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_09.pdf&ved=0ahUKEwitj8nH0KnRAhUC7GMKHYxoCWoQFghOMA8&usg=AFQjCNHiSV17-hjfnjrbqPgfk1uzd8Jmqw&sig2=qNjM2E10sGt4RPiKRgWz4A

remember

[latex][\hat{L_x},\hat{L_y}]=i\hbar \hat{L_z}[/latex],

[latex][\hat{L_y},\hat{L_z}]=i\hbar \hat{L_x}[/latex]

[latex][\hat{L_z},\hat{L_x}]=i\hbar \hat{L_y}[/latex]

 

these are components of a vector on a cartesian coordinate system

[latex]L_x=\frac{\hbar}{\sqrt{2}}\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}[/latex]
[latex]L_y=\frac{\hbar}{\sqrt{2i}}\begin{pmatrix}0&1&0\\-1&0&-1\\0&-1&0\end{pmatrix}[/latex]
[latex]L_z=\hbar\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}[/latex]

[latex]\langle\psi|L_x|\psi\rangle=(\psi_1^*,\psi_2^*,\psi_3^*)\frac{\hbar}{\sqrt{2}}\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}\begin{pmatrix}\psi_1\\ \psi_2\\ \psi_3\end{pmatrix}[/latex]
[latex]=\frac{\sqrt{\hbar}}{2}(\psi_1^*\psi_2^*\psi_3^*)\begin{pmatrix}\psi_2\\ \psi_1+\psi_3\\\psi_2\end{pmatrix}[/latex]
[latex]=\frac{\sqrt{\hbar}}{2}(\psi_1^*\psi_2\psi_2^*(\psi_1+\psi_2)+\psi_3^*\psi_2)[/latex]
I'm curious how much work have you done with tensors? this seems to be where your having difficulties. Though I could be wrong on that



Though if your missing the details below which I should have started with the above makes more sense

[latex]\overrightarrow{L}=\overrightarrow {r}*\overrightarrow{p}=\begin{pmatrix}i&j&k\\x&y&z\\p_x&p_y&p_z\end{pmatrix}=i(yp_z-zp_y)+j(zp_x-xp_z)+k (xp_y-yp_x)[/latex]

 

the magnitude of each component being

[latex]L_x=yp_z-zp_y[/latex]

[latex]L_y=yp_x-zp_z[/latex]

[latex]L_z=yp_y-zp_x[/latex]

 

hope this helps

Edited January 5, 2017 by Mordred

[Mordred]

edit bad post

I'm having a little trouble with some commutational relations between different components of the angular momentum operators. I'm OK up to:

 

[math][\hat{L_x},\hat{L_y}]=\hat{Y}\hat{P_z}\hat{Z}\hat{P_x} - \hat{Z}\hat{P_x}\hat{Y}\hat{P_z} + \hat{Z}\hat{P_y}\hat{X}\hat{P_z} - \hat{X}\hat{P_z}\hat{Z}\hat{P_y}[/math]

 

and the apparently this is equal to

 

[math]\hat{Y}\hat{P_x}(\hat{P_z}\hat{Z} - \hat{Z}\hat{P_z}) + \hat{X}\hat{P_y}(\hat{Z}\hat{P_z} - \hat{P_z}\hat{Z})[/math]

 

I don't understand this 'factoring out' of operators. I thought it was a typo in my notes and the answer is:

 

[math]\hat{Y}\hat{P_z}(\hat{Z}\hat{P_z} - \hat{Z}\hat{P_z}) + \hat{X}\hat{P_y}(\hat{Z}\hat{P_z} - \hat{Z}\hat{P_z})[/math]

 

but then the rest of the derivation doesn't work.

 

I think i'm missing some property of commutators, but can't see what it is.

 

Help appreciated.

Just re-read this let me think about it.

Edited January 5, 2017 by Mordred

[uncool]

I'm having a little trouble with some commutational relations between different components of the angular momentum operators. I'm OK up to:

 

[math][\hat{L_x},\hat{L_y}]=\hat{Y}\hat{P_z}\hat{Z}\hat{P_x} - \hat{Z}\hat{P_x}\hat{Y}\hat{P_z} + \hat{Z}\hat{P_y}\hat{X}\hat{P_z} - \hat{X}\hat{P_z}\hat{Z}\hat{P_y}[/math]

 

and the apparently this is equal to

 

[math]\hat{Y}\hat{P_x}(\hat{P_z}\hat{Z} - \hat{Z}\hat{P_z}) + \hat{X}\hat{P_y}(\hat{Z}\hat{P_z} - \hat{P_z}\hat{Z})[/math]

 

I don't understand this 'factoring out' of operators. I thought it was a typo in my notes and the answer is:

 

[math]\hat{Y}\hat{P_z}(\hat{Z}\hat{P_z} - \hat{Z}\hat{P_z}) + \hat{X}\hat{P_y}(\hat{Z}\hat{P_z} - \hat{Z}\hat{P_z})[/math]

 

but then the rest of the derivation doesn't work.

 

I think i'm missing some property of commutators, but can't see what it is.

 

Help appreciated.

 

 

Remember some assumptions on the commutation relations of position and momentum - position and momentum operators with different indices (e.g. [math]\hat{Y},\hat{P_z}[/math]) commute, while ones with the same index (e.g. [math]\hat{Z},\hat{P_z}[/math]) do not. In your expression, the only terms with common indices deal with Z, so you can feel free to move the others in any way you want - including, as here, to the front. So when you do that for the first term, you get [math]\hat{Y}\hat{P_z}\hat{Z}\hat{P_x} = \hat{Y}\hat{P_x}\hat{P_z}\hat{Z}[/math]. However, you can't then switch [math]\hat{P_z}[/math] and [math]\hat{Z}[/math], as they do not commute - so this is the simplest you can get. Similarly, the second term becomes [math]\hat{P_x}\hat{Y}\hat{Z}\hat{P_z}[/math].

 

Does that help?

[Prometheus]

I'm curious how much work have you done with tensors? this seems to be where your having difficulties. Though I could be wrong on that

 

Absolutely none, and it's not coming up either.

 

 

Does that help?

 

Ah, yes. I did know this, but didn't apply it. I'll try later and let you know, but i think i'll be OK now.

 

Yep, that worked a treat, thanks.

Edited January 5, 2017 by Prometheus

[Mordred]

k good glad you got it. By the time I reread the OP it was too late to think clearly lol. Glad uncool covered it.

 

Good to know on the matrix/tensors not being on your curriculum. It will help better constrain future replies

Edited January 5, 2017 by Mordred