Basis states, matrix elements and angular momentum

[Onamor]

Homework Statement

The last 2 parts of the attached photo. (4 and 6 marks)
Im really not sure how to go about them in a (clever) way that won't take 2 hours.

Homework Equations

Possibly the fact that the product of the raising/lowering operators, J_-J₊ = J²_x + J²_y

Answers to previous question:
the matrix of J² = 15/4 $$\hbar$$² I₄ (the identity)

and J_z is $$\hbar$$/2 times
[3 0 0 0]
[0 1 0 0]
[0 0-1 0]
[0 0 0-3]

The Attempt at a Solution

For the explanation you could say that any operator represented in terms of its basis states is diagonal - but then how can you tell that those four given eigenvectors are the basis states of H? (you're only told that they are eigenvectors of Jz).

The eigenvalues are the enegry eigenvalues (along the diagonal), but to find the matrix (knowing its diagonal) you can just find <1|H|1>, <2|H|2>, <3|H|3> and <4|H|4>. But you need some sort of matrix or equation for the J²_x + J²_y part of the Hamiltonian - its possible to find their matrices in this basis but I just can't believe there isn't an easier way for 6 marks...

Thanks again to anyone who can help

 

[fzero]

For the explanation you could say that any operator represented in terms of its basis states is diagonal - but then how can you tell that those four given eigenvectors are the basis states of H? (you're only told that they are eigenvectors of Jz).

Did you compute $$[J^2,J_z]$$ for part 2? What does it tell you about the eigenvectors for $$J^2$$ and $$J_z$$?

The eigenvalues are the enegry eigenvalues (along the diagonal), but to find the matrix (knowing its diagonal) you can just find <1|H|1>, <2|H|2>, <3|H|3> and <4|H|4>. But you need some sort of matrix or equation for the J²_x + J²_y part of the Hamiltonian - its possible to find their matrices in this basis but I just can't believe there isn't an easier way for 6 marks...

Can you write $$J_x^2 +J_y^2$$ in terms of $$J^2$$ and $$J_z$$?

 

[Onamor]

Hi, thanks so much for your help!

Did you compute $$[J^2,J_z]$$ for part 2? What does it tell you about the eigenvectors for $$J^2$$ and $$J_z$$?

Yes, their commutator is zero, so they commute and it is possible to find simultaneous eigenfunctions of both of them.

Can you write $$J_x^2 +J_y^2$$ in terms of $$J^2$$ and $$J_z$$?

I think $$J^2 - J_z^2$$ should do it?

Since (given your first tip) an eigenfunction of $$J_z$$ is simultaneously an eigenfunction of $$J^2$$, would rewriting the Hamiltonian using this show that it is expressable in terms of the given four basis states? - and therefore is diagonal?

And then I guess to find the Hamiltonian's (diagonal) matrix elements you can now just use the previously calculated matrices for $$J^2$$ and $$J_z$$?

That would seems a lot more sensible than what I attempted...

 

[fzero]

Since (given your first tip) an eigenfunction of $$J_z$$ is simultaneously an eigenfunction of $$J^2$$, would rewriting the Hamiltonian using this show that it is expressable in terms of the given four basis states? - and therefore is diagonal?

And then I guess to find the Hamiltonian's (diagonal) matrix elements you can now just use the previously calculated matrices for $$J^2$$ and $$J_z$$?

That would seems a lot more sensible than what I attempted...

Yes, just compute a bit and you'll find that things are as you say. It's hard to get all of this straight just from lectures, so it's exercises like this that really teach you how things work.

 

[Onamor]

Thanks again for your help. Yes, all maths needs practise, but QM is nearly unlearnable from books and lectures alone.