Requesting guidance on Commutators & Intro QM

AI Thread Summary
The discussion centers on a request for guidance on solving a quantum mechanics problem involving commutators, specifically calculating [Alpha, Beta] given certain definitions of Alpha and Beta. The original poster expresses uncertainty about their approach and seeks clarification on the use of angular momentum and the implications of the Plus sign in the Beta term. A response advises treating Alpha and Beta as operators rather than simple quantities, emphasizing the importance of understanding fundamental commutation relationships. Additionally, the responder suggests using LaTeX for clearer communication in future posts. The overall focus is on enhancing conceptual understanding of quantum mechanics through collaborative learning.
warhammer
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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 dependant on samaritans like you and forums like these to fine tune my conceptual knowledge 🙏🏻

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warhammer said:
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/.
 
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