Let $\vec{p} = p_x \hat{x} + p_y \hat{y} + p_z \hat{z}$, and also use the notation $|\vec{p}| = p$, where $p^2 = p_x^2 + p_y^2 + p_z^2$. What is the difference between the operator $\hat{p}^2 = \hat{\vec{p}} \cdot \hat{\vec{p}}$, and the operator $\widehat{p^2}$? And which one is the operator tha...

Are electrons created from a battery and then 'injected' into the copper wire? If no, then does that mean that the battery looses power because the copper wire has run out of free electrons?