Quantum Mechanics - Matrix representations

[Graham87]

Homework Statement

See pic

Relevant Equations

See pic

I have found J^2 and Jz, but I am not sure how to find Jx and Jy.
I’m thinking maybe use J+-=Jx+-iJy ? But I get unclear results.

Thanks!

 

[PeroK]

Homework Statement:: See pic
Relevant Equations:: See pic

View attachment 313615
I have found J^2 and Jz, but I am not sure how to find Jx and Jy.
I’m thinking maybe use J+-=Jx+-iJy ? But I get unclear results.

View attachment 313617

Thanks!

Using $J_{\pm}$ sounds like a good idea. Show us what you get.

 

[DrClaude]

I’m thinking maybe use J+-=Jx+-iJy ? But I get unclear results.

You have to do it the other way around: Express $J_x$ and $J_y$ in terms of $J_+$ and $J_-$.

 

[Graham87]

I got this, but it gets messy when I try to find J+- with this expression:

I don’t know how to form the matrix, what goes where.

 

[DrClaude]

Matrix elements are found using $\braket{j',m' | J_x | j, m}$, so you can start by calculating $J_+ \ket{j,m}$ and $J_- \ket{j,m}$ explicitly for the 4 $\ket{j,m}$ states you need to consider here.

 

[Graham87]

Something like this?

 

[PeroK]

Good effort, but I don't agree with some of the entries in the $J_{\pm}$ matrices.

 

[PeroK]

Here's a tip. A lot of physics textbooks write one formula with various $\pm$ and $\mp$. I find it easier to keep the formulas separate with either $+$ or $-$.

 

[PeroK]

Where in the matrix did you not agree ?

The non-zero entries!

 

[Graham87]

The non-zero entries!

Thanks! Just noticed my error.
cheers!