Show that the Hamiltonian commutes with Angular momentum

[tarkin2]

Homework Statement

[/B]
Parts (c) and (f) are the ones I'm having trouble with;

Homework Equations

The Attempt at a Solution

[/B]
For (c), I assume the problem is meant to involve using the result from part (b), which was H = g(J² - L² - S²)/2 .

I was trying just to do it by first showing that H commutes with J² , and then was going to do the same for L² and S², and also I would have just stated that Jz commutes with J² and therefore also with the Hamiltonian.

But I wasn't entirely sure how to show that this. I tried:

[H, J²] = (g/2) [ J² - L² - S² , J² ]
=(g/2) ([ J² , J²] - [L², J² ] - [S² , J² ])
= (g/2) (0- [L², J² ] - [S² , J² ])

But I wasn't sure how to show that these last 2 terms are 0. Also I assume this isn't the best method, because it doing it this way you would be answering part (f) in the process. Is there a different method, for only 3 marks?

Any help is appreciated

 

[MathematicalPhysicist]

If I remember correctly a scalar operator commutes with every operator, and a square of an operator is scalar.

 

[MathematicalPhysicist]

And a scalar operator is basically, $s\cdot I$ where $s$ is some scalar and $I$ is the identity operator, and you can prove that the identity operator commutes with any other operator.

 

[nrqed]

Homework Statement

[/B]
Parts (c) and (f) are the ones I'm having trouble with;

View attachment 224710

Homework Equations

The Attempt at a Solution

[/B]
For (c), I assume the problem is meant to involve using the result from part (b), which was H = g(J² - L² - S²)/2 .

I was trying just to do it by first showing that H commutes with J² , and then was going to do the same for L² and S², and also I would have just stated that Jz commutes with J² and therefore also with the Hamiltonian.

But I wasn't entirely sure how to show that this. I tried:

[H, J²] = (g/2) [ J² - L² - S² , J² ]
=(g/2) ([ J² , J²] - [L², J² ] - [S² , J² ])
= (g/2) (0- [L², J² ] - [S² , J² ])

But I wasn't sure how to show that these last 2 terms are 0. Also I assume this isn't the best method, because it doing it this way you would be answering part (f) in the process. Is there a different method, for only 3 marks?

Any help is appreciated

Hi. There is no need to rewrite the Hamiltonian, it is easier to leave in the form $g L \cdot s$. First, note that $L^2$ is a Casimir operator, it commutes with any function of the $L_i$. They give a hint: what is the commutation relations between the $L_i$ and $s_j$? Using these two results it is trivial to calculate the commutator or $L^2$ with the Hamiltonian. The other commutators are easy to find using again that $s^2$ commutes with all the $s_i$ and that $J^2$ commutes with all the $J_i$.