Total Angular Momentum Commutation Relations for 2 Particle

[Sekonda]

Hey,

I'm not exactly sure how much this question wants, however the two in question are parts a) and b) below.

So part a) asks to write the expression for the total angular momentum J, I though this was just:

$$\hat{J}=\hat{J}^{(1)}+\hat{J}^{(2)}$$

but when we come to showing it satisfies similar commutation relations - I'm not really sure how to do this, I was thinking of expanding J like so

$$\hat{J}=\hat{J}_{x}+\hat{J}_{y}+\hat{J}_{z}=(\hat{J}^{(1)}_{x}+\hat{J}^{(2)}_{x})+(\hat{J}^{(1)}_{y}+\hat{J}^{(2)}_{y})+(\hat{J}^{(1)}_{z}+\hat{J}^{(2)}_{z})$$

but wasn't sure if this was the correct route and if the last expansion or even the first was necessary.

Part b I think relies on the commutation relations which we find in a) - I'm fairly sure I can show that they all commute but I don't know what expansion/what other commutation relation I need to derive in order to show these commutators are conserved.

Thanks for any help!
SK

 

[G01]

Hey,

I'm not exactly sure how much this question wants, however the two in question are parts a) and b) below.

So part a) asks to write the expression for the total angular momentum J, I though this was just:

$$\hat{J}=\hat{J}^{(1)}+\hat{J}^{(2)}$$

but when we come to showing it satisfies similar commutation relations - I'm not really sure how to do this, I was thinking of expanding J like so

$$\hat{J}=\hat{J}_{x}+\hat{J}_{y}+\hat{J}_{z}=(\hat{J}^{(1)}_{x}+\hat{J}^{(2)}_{x})+(\hat{J}^{(1)}_{y}+\hat{J}^{(2)}_{y})+(\hat{J}^{(1)}_{z}+\hat{J}^{(2)}_{z})$$

but wasn't sure if this was the correct route and if the last expansion or even the first was necessary.

Part b I think relies on the commutation relations which we find in a) - I'm fairly sure I can show that they all commute but I don't know what expansion/what other commutation relation I need to derive in order to show these commutators are conserved.

Thanks for any help!
SK

You're on the right track. Define the total angular momentum vector as you did, and the individual components in a similar way. You should then be able to explicitly find the commutators by working them out explicitly in terms of the single particle operators and the commutation relations given.

 

[Sekonda]

Ahh good to know, will give it a try now and get back if I'm having trouble!

 

[Sekonda]

I managed to attain this commutation relation:

$$[\hat{J}^{(1)},\hat{J}^{(2)}]=0$$

and

$$[\hat{J},\hat{J}^{(1/2)}]=0$$

but that's all, I think those are all I need?

 

[vela]

You are keeping in mind that $\hat{\vec{J}}$ is a vector, right? The commutation relations involve individual components of this vector and the (non-vector) operator $\hat{J}^2$.

 

[Sekonda]

Though I seem to be having trouble showing

$$[\hat{J},\hat{J}_{z}]=0$$

 

[Sekonda]

Ok so the operator

$$\hat{J}^{2}$$

is equal to what? The individual components squared and then summed? Or the sum of the individual components squared?

 

[G01]

Ok so the operator

$$\hat{J}^{2}$$

is equal to what? The individual components squared and then summed? Or the sum of the individual components squared?

The J operators are vector operators:

$$\hat{J}^2=\hat{\vec{J}}\cdot\hat{\vec{J}}$$

What is the dot product of two vectors in terms of the component vectors?

 

[vela]

Ok so the operator

$$\hat{J}^{2}$$

is equal to what? The individual components squared and then summed? Or the sum of the individual components squared?

That's really something you should know or at least look up in your textbook before asking here.

 

[Sekonda]

No worry I've managed to do it now, was getting confused over the J^2 operator...

Cheers guys!
SK