(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The canonical commutation relations for a particl moving in 3D are

[tex] [\hat{x},\hat{p_{x}}]= i\hbar [/tex]

[tex] [\hat{y},\hat{p_{y}}]= i\hbar [/tex]

[tex] [\hat{z},\hat{p_{z}}]= i\hbar [/tex]

and all other commutators involving x, px, y ,py, z , pz (they should all have a hat on eahc of them signifying that htey are operators) are zero. These relations can be used to show that the operators for the orbital angular mometum obey the following commutation relations

[tex] [\hat{L_{x}},\hat{L_{y}}]= i\hbar \hat{L_{z}} [/tex]

[tex] [\hat{L_{y}},\hat{L_{z}}]= i\hbar \hat{L_{x}} [/tex]

[tex] [\hat{L_{z}},\hat{L_{x}}]= i\hbar \hat{L_{y}} [/tex]

Using

[tex] \hat{L_{x}} = \hat{y}\hat{p_{z}} - \hat{z}\hat{p_{y}} [/tex]

[tex] \hat{L_{y}} = \hat{z}\hat{p_{x}} - \hat{x}\hat{p_{z}} [/tex]

Verify that

[tex] [\hat{L_{x}},\hat{L_{y}}] = [\hat{y}\hat{p_{z}},\hat{z}\hat{p_{x}}]+[\hat{z}\hat{p_{y}},\hat{x}\hat{p_{z}}] [/tex]

3. The attempt at a solution

I tried opening up the commutators and it really did get me anywhere

here is what i did

[tex] [\hat{y}\hat{p_{z}},\hat{z}\hat{p_{x}}]+[\hat{z}\hat{p_{y}},\hat{x}\hat{p_{z}}] = yp_{y}zp_{x} - zp_{x}yp_{z} + zp_{y}xp_{z} - xp_{z}zp_{y} [/tex]

and the left hand side yields

[tex] yp_{z}zp_{x} - yp_{z}xp_{z} - zp_{y}zp_{x} + zp_{y}xp_{z} + zp_{x} yp_{z} - zp_{x} zp_{y} - xp_{z}yp_{z} + xp_{z} z p_{y} [/tex]

nothing seems to simplify... or is there something im missing...?

o and i did not put hats on eahc of them because it would just too much typing...

thanks for your help!

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# Canonical commutation relations for a particle

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