[Physics] Quantum Mechanics Lx operator

quantum mechanics

Show that if the state $ \rvert\gamma\rangle $ is real, then the expectation value of each component of the angular momentum is zero. Does this imply the angular momentum is zero?

My Work:

$$ \langle\ \gamma\rvert l_x\rvert\gamma\rangle =\langle\ \gamma\rvert l_x\rvert\gamma\rangle^* =\langle\ \gamma\rvert l_x^*\rvert\gamma\rangle$$
$$l_x=\frac12(l_+ + l_-)$$
$$l_-^+= l_x + _-^+il_y$$

Best Answer

Try to answer the following question as a warm-up. What can you say about the expectation value of the $1D$ momentum operator $p = -i \hbar \tfrac{d}{dx}$ of a real wavefunction? Note that the expectation value itself should be real, because momentum is a real quantity (and the operator is hermitian).