# Calculation with Pauli matrices

1. Sep 2, 2016

### frerk

1. The problem statement, all variables and given/known data

Hey :-)
I just need some help for a short calculation.
I have to show, that
$$(\sigma \cdot a)(\sigma \cdot b) = (a \cdot b) + i \sigma \cdot (a \times b)$$

3. The attempt at a solution

I am quiet sure, that my mistake is on the right side, so I will show you my calculation for this one:
$$a_xb_x + a_yb_y+a_zb_z + i\sigma_x (a_yb_z - a_3b_2) + i\sigma_y (a_zb_x-a_xb_z) + i\sigma_z (a_xb_y -a_yb_x)$$

The last 3 terms are a 2x2 matrix and the first 3 terms are just a scalar...

would be happy fora small hint what is wrong :-)
Thank you

2. Sep 2, 2016

### blue_leaf77

There is actually an identity matrix to be multiplied wit $a\cdot b$.

3. Sep 2, 2016

### frerk

Yes, right. that brings to to the result I want.
Is there a rule, why I have to multiply the result of the dot product with the idendity matrix?
Because the other terms include a Pauli Matrix and the result
of the dot produkt must adapt to that structure?

4. Sep 2, 2016

### blue_leaf77

Of course it can be proven using the more fundamental properties of Pauli matrices, especially their commutation and anti-commutation. An easy prove can be found here.