Help - Pauli Spin Matrices Proof

[Fjolvar]

Hello, I attached a copy of the problem and my attempted solution. The three Pauli spin matrices are given above the problem. I'm having trouble getting the right side to equal the left side, so I'm assuming I'm doing something wrong. When I got towards the end it just wasn't looking right. Any help would be greatly appreciated, even if you can just point out my mistake. Thank you in advance!

 

[Fjolvar]

Did I use the identity matrix wrong? It says its a 2x2 matrix but the sigma matrix has 3 dimensions...

 

[vela]

You can do that way, but it's a lot tidier if you use the properties of the Pauli matrices that were established in the problem right above the one you're trying to do now. Are you familiar with the Levi-Civita symbol?

 

[Fjolvar]

I am somewhat familiar with Levi Civita since we covered it briefly. I finished the problem the long way, but I'm interested in learning how to use the Levi Civita symbol.

 

[vela]

You can combine properties (b) and (c) in the previous problem to show that

$$[\sigma_j,\sigma_k] = 2i\varepsilon_{jkl}\sigma_l$$

which is just the regular commutation relation for angular momentum written in terms of the Pauli matrices. Also, you need to know that the cross product of two vectors can be expressed as

$$(\vec{a} \times \vec{b})_k = \varepsilon_{ijk}a_i b_j$$

in terms of the Levi-Civita symbol.

Using implied summation notation, you can write the lefthand side as

$$(a_j\sigma_j)(b_k\sigma_k) = a_j b_k \sigma_j\sigma_k$$

Use the commutation relation to switch the order of the Pauli matrices on the RHS, and then use property (c) from the previous problem to switch the order in the remaining product back.