Requesting guidance on Commutators & Intro QM

In summary, the person is asking how to solve a problem using commutation relationships, and provides a link to a video that can help.
  • #1
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 🙏🏻

IMG20220605141136__01.jpg
 
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  • #2
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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1. What is a commutator in quantum mechanics?

A commutator is a mathematical operation that measures the difference between two operators. In quantum mechanics, it is used to determine the uncertainty between two observables, such as position and momentum.

2. How is a commutator calculated?

A commutator is calculated by taking the product of two operators and then subtracting the product of the same two operators in reverse order. This can be represented by the equation [A, B] = AB - BA.

3. What is the significance of commutators in quantum mechanics?

Commutators are significant in quantum mechanics because they help to determine the uncertainty between two observables. They also play a crucial role in the Heisenberg uncertainty principle, which states that the more precisely one observable is known, the less precisely the other can be known.

4. How are commutators related to the wave function?

The commutator of two operators is related to the wave function through the Heisenberg uncertainty principle. The commutator of two operators is equal to the expectation value of the commutator of the corresponding wave functions.

5. How can I use commutators to solve problems in quantum mechanics?

Commutators can be used to solve problems in quantum mechanics by helping to determine the uncertainty between two observables. They can also be used to calculate the expectation values of observables and to derive equations of motion for quantum systems.

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