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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 $V(x) = (m\omega^2/2)x^2$?

          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 $V(x) = (m\omega^2/2)x^2$?

Problem 1: Generalized uncertainty principle
Consider a particle in a time-independent potential V(x). The Hamiltonian of the system is then,
Ĥ = (p̂^2)/(2m) + V(x)
a. Using the relation between force and momentum, F = dp/dt, find an expression for the force
operator, F̂.
b. Find the force-momentum uncertainty relation (i.e., ≥ ?) 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) = (mω^2/2)x^2?

Added by Suzanne M.

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