- #1

- 668

- 68

## Homework Statement

Verify the following commutation relations using [tex] \vec J = \vec Q \times \vec p[/tex] and [tex][Q_{\alpha},p_{\beta}]=i \delta_{\alpha \beta} I[/tex]

1. [tex] [J_{\alpha}, J_{\beta}]=i \epsilon_{\alpha \beta \gamma} J_{\gamma}[/tex]

2. [tex] [J_{\alpha}, p_{\beta}]=i \epsilon_{\alpha \beta \gamma} p_{\gamma}[/tex]

3. [tex] [J_{\alpha}, G_{\beta}]=i \epsilon_{\alpha \beta \gamma} G_{\gamma}[/tex]

## Homework Equations

note epsilon is 1 when alpha beta gamma are in permutable order, -1 when they are not, and 0 if any are equal.

## The Attempt at a Solution

Diving right in on the first one,

[tex] [J_{\alpha},J_{\beta}]=[(Q \times p)_{\alpha}, (Q \times p)_{\beta}] = (Q \times p)_{\alpha}(Q \times p)_{\beta}-(Q \times p)_{\beta}(Q \times p)_{\alpha}=? [/tex]

is this the right way to go about this? should i be using the jacobi identity instead?