- #1

Kamikaze_951

- 7

- 0

## Homework Statement

To whom it may concern,

I am trying to understand the central force problem of the Dirac equation. In particular, I am following Sakurai's Advanced Quantum Mechanics book. There (section 3.8, p.122), it is shown that there is an operator

[itex]K = \beta(\Sigma . L + \overline{h})[/itex]

where [itex]\Sigma_i = \left(

\begin{array}{cc}

\sigma_i && 0 \\

0 && \sigma_i \\

\end{array}\right)[/itex]

This operator commutes with the hamiltonian H and the total angular momentum J. At the top of page 123, it is shown that [itex]K^2 = J^2 + \overline{h}/4[/itex]. At some point, we get

[itex]K^2 = L^2 + i\Sigma . (L × L) + 2\overline{h}\Sigma . L + \overline{h} = L^2 + \overline{h}\Sigma . L + \overline{h}^2[/itex]

I don't understand this last equality. Why is [itex]i \Sigma . (L × L) = -\overline{h}\Sigma.L [/itex]?

## Homework Equations

## The Attempt at a Solution

I looked a lot in the literature, but I didn't find any more precise calculation. I don't seem to grasp what L × L means. I know that [itex]\vec{L} = \vec{x}× (-i\overline{h}\nabla)[/itex]. If we had vectors instead of operators, we would have [itex]\vec{L} × \vec{L} = 0[/itex], but what does the vector product of 2 identical operators mean?

Thanks for the help.

Kami