Recent content by B L

That's the thing: when I asked my TA, he said that we are not to assume the summation convention. That's the source of my confusion - I have no idea what this question even means without the summation convention. (I'm literally the only person in the course who knows what a tensor is).

Homework Statement Using the position space representation, prove that: \left[L_i, x_j\right] = i\hbar\epsilon_{ijk}x_k . Similarly for \left[L_i, p_j\right] . Homework Equations Presumably, L_i = \epsilon_{ijk}x_jp_k . \left[x_i, p_j\right] = i\hbar\delta_{ij} . The Attempt at a...

I know why I went wrong - what I didn't realize is that by differentiation wrt c, I really meant partial differentiation wrt c^ij (and differentiation of c^ij wrt s) As soon as you do that the problem is trivial.

Am I correct in thinking that the corresponding tangent vector (i.e. element of the Lie algebra so(3) \cong T_eSO(3) ) would be -\frac{\partial}{\partial x^{12}} +\frac{\partial}{\partial x^{21}} ? Thanks for the help.

Homework Statement Let c(s) = \left( \begin{array}{ccc} \cos(s) & -\sin(s) & 0 \\ \sin(s) & \cos(s) & 0 \\ 0 & 0 & 1 \end{array} \right) be a curve in SO(3). Find the tangent vector to this curve at I_3 . Homework Equations Presumably, the definition of a tangent vector as a differential...

I don't think we're supposed to use eigenvectors, but I'll give that a shot, thanks! Any other ideas? I'm way too stumped given how seemingly simple this thing is (especially compared to the rest of the assignment).

Homework Statement Let M be a differentiable manifold, p \in M. Suppose A \in T_{1,p}^1(M) is symmetric with respect to its indices (i.e. A^i_j = A^j_i) with respect to every basis. Show that A^i_j = \lambda \delta^i_j, where \lambda \in \mathbb{R}. Homework Equations The Attempt at a...