Prometheus Posted January 4, 2017 Share Posted January 4, 2017 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. Link to comment Share on other sites More sharing options...
Mordred Posted January 4, 2017 Share Posted January 4, 2017 (edited) 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=p0G0kk7RP3Dyd8tlwjGq7whttp://farside.ph.utexas.edu/teaching/qmech/Quantum/node71.htmlActually 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=qNjM2E10sGt4RPiKRgWz4Aremember [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 thatThough 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 Link to comment Share on other sites More sharing options...
Mordred Posted January 5, 2017 Share Posted January 5, 2017 (edited) 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 Link to comment Share on other sites More sharing options...
uncool Posted January 5, 2017 Share Posted January 5, 2017 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? 2 Link to comment Share on other sites More sharing options...
Prometheus Posted January 5, 2017 Author Share Posted January 5, 2017 (edited) 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 Link to comment Share on other sites More sharing options...
Mordred Posted January 5, 2017 Share Posted January 5, 2017 (edited) 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 1 Link to comment Share on other sites More sharing options...
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