[SHQ/QM-07003] Short Questions ---- Commutator

Write short answers to the following questions.

1. If two dynamical variable $A,B$ , $\Delta A \Delta B \ge \text{minimum value}$, what is this minimum value related to?
2. If $A=L_x$, what could an operator $B$ having an uncertainty relation with $L_x$?
3. Name a thought experiment which has shows that an increase in accuracy of position measurement results in increase in uncertainty in the momentum.
4. Name a thought experiment which has shows that an attempt to increase in accuracy of momentum measurement results in increase in uncertainty in the position.
5. Although $L_x$ and $L_y$ do not commute, in special states they can have a very accurate simultaneous value. What this state is ? Why this does not contradict the general statement?
6. If uncertainty $\Delta X$ of a dynamical variable $\hat{X}$ is zero in a state $\psi$, then what is the special relationship the operator $\hat{X}$ and the state $\psi$ must have?
7. State the mathematical property that two dynamical variables must have if they do not have an uncertainty relation for their simultaneous measurement..
8. Identify the dynamical variable $X$ having the property that for every $Y$ having no uncertainty relation with $X$, the average value of $Y$ does not change with time.
9. Classically a particle stays at rest at the bottom of potential well, but not a quantum particle? What explains this essential quantum phenomenon?
10. That an electron cannot be bound inside a nucleus can be understood by appealing to a famous principle. Identify this principle.