- #1
B L
- 8
- 0
Homework Statement
Using the position space representation, prove that:
[itex] \left[L_i, x_j\right] = i\hbar\epsilon_{ijk}x_k [/itex].
Similarly for [itex] \left[L_i, p_j\right] [/itex].
Homework Equations
Presumably, [itex] L_i = \epsilon_{ijk}x_jp_k [/itex].
[itex] \left[x_i, p_j\right] = i\hbar\delta_{ij} [/itex].
The Attempt at a Solution
[itex] \left[L_i, x_j\right] = \epsilon_{ijk}\left[x_jp_k, x_j\right]
= \epsilon_{ijk}\left(x_jp_kx_j - x_jx_jp_k\right)
= \epsilon_{ijk}x_j\left(p_kx_j - x_jp_k\right)
= \epsilon_{ijk}x_j\left[p_k, x_j\right]
= -i\hbar\epsilon_{ijk}x_j\delta_{jk} [/itex]
which is where I become confused - it seems to me that the right hand side is always zero (if the Kronecker delta is nonzero, the Levi-Civita symbol is zero, and vice-versa).
Any help is much appreciated.