Requesting guidance on Commutators & Intro QM

[warhammer]

Homework Statement

If Alpha=i( x*P(y) - y*P(x) ) & Beta=i( y*P(z) + z*P(y) ) are given, find [Alpha, Beta]

Relevant Equations

[Alpha, Beta]= αβ - βα

I have approached this question step by step as shown in the image attached.

I request someone to please guide if I have approached the (incomplete) solution correctly and also guide towards the complete solution, by helping me to rectify any mistakes I may have made.

I'm still unsure how to proceed here. Someone also suggested to use it in form of Angular Momentum, but what about the Plus sign in the Beta term, since Lx is specified as yPz-zPy !

PS: Please bear with me patiently. I had a horrible Prof this sem who shot my confidence in the subject to bits having me to learn all of QM in self study mode. Therefore I'm dependent on samaritans like you and forums like these to fine tune my conceptual knowledge

 

[Steve4Physics]

Homework Statement:: If Alpha=i( x*P(y) - y*P(x) ) & Beta=i( y*P(z) + z*P(y) ) are given, find [Alpha, Beta]
Relevant Equations:: [Alpha, Beta]= αβ - βα

Hi @warhammer. A few of points...

EDITed (mainly corrections as I forget about the 'z's)

It looks like you may be thinking of $\alpha, \beta, x, y, z, p_x, p_y$ and $p_z$ as simple quantities (real or complex values). In this case $[\alpha, \beta]$ would necessarily equal zero. (Why?)

Presumably you are intended to treat them as operators. They would generally be written with ‘hats’: $\hat {\alpha}, \hat {\beta}, \hat x$ etc.

Before you tackle this question, you should understand how the (hopefully familiar) commutation relationship $[\hat x, \hat {p_x}] = iℏ$ is derived. Try this video for example:

Once that’s clear, you should be in a better position to answer your original question.

If you intend posting here regularly, you are advised to use LaTeX to write equations; this makes it a lot easier to read your posts (and is a useful skill anyway). For example see https://www.physicsforums.com/help/latexhelp/.