Understanding the Determinant of Commutator Matrices in Angular Momentum Systems

[ognik]

Hi, I've just wierded myself out so time to stop for today, but afore I go ...

Angular momentum matrices (also Pauli) anti-commute as $ [J_x, J_y] = iJ_z $

So $ Det\left( [J_x, J_y] \right) = i Det(J_z) $
$\therefore Det(J_xJ_y)-Det(J_yJ_x) = i Det(J_z) $
$\therefore Det(J_x)Det(J_y)-Det(J_y)Det(J_x) = i Det(J_z) $

I think you can see why I am confused here, usual question (sigh) what have I done wrong please?

 

[topsquark]

Hi, I've just wierded myself out so time to stop for today, but afore I go ...

Angular momentum matrices (also Pauli) anti-commute as $ [J_x, J_y] = iJ_z $

So $ Det\left( [J_x, J_y] \right) = i Det(J_z) $
$\therefore Det(J_xJ_y)-Det(J_yJ_x) = i Det(J_z) $
$\therefore Det(J_x)Det(J_y)-Det(J_y)Det(J_x) = i Det(J_z) $

I think you can see why I am confused here, usual question (sigh) what have I done wrong please?

\(\displaystyle |x + y| \neq |x| + |y| \)

-Dan

 

[Ackbach]

Also,
$$\text{det}(iJ_z)\not=i \, \text{det}(J_z),\qquad \text{but} \qquad \text{det}(iJ_z)=i^n \, \text{det}(J_z),$$
where $n\times n$ is the size of the $J_z$ matrix.

 

[ognik]

\(\displaystyle |x + y| \neq |x| + |y| \)

-Dan

Sorry, don't understand, how does relate to commutator? Ta

 

[Ackbach]

Sorry, don't understand, how does relate to commutator? Ta

I could be wrong, but I think topsquark is saying that going from the first to the second line is invalid. That is,
$$\text{det}(AB-BA) \not= \text{det}(AB) - \text{det}(BA).$$

 

[ognik]

Sometimes I can't see the wood for the trees, thanks guys.