Problem 1: Generalized uncertainty principle
Consider a particle in a time-independent potential $V(x)$. The Hamiltonian of the system is then,
$\hat{H} = \frac{\hat{p}^2}{2m} + V(x)$
a. Using the relation between force and momentum, $F = dp/dt$, find an expression for the force
operator, $\hat{F}$.
b. Find the force-momentum uncertainty relation (i.e., $\sigma_F \sigma_p \ge ?$) in terms of $V(x)$.
c. What is the force-momentum uncertainty relation for the case of a 1D quantum harmonic
oscillator with $V(x) = (m\omega^2/2)x^2$?