Discrepancy in uncertainty among various operators

[hokhani]

TL;DR

Momentum uncertainty isn't identical for position and Hamiltonian.

Consider a particle in an eigenstate of the Hamiltonian when the particle is in a 1D box with very large length $L$. The operators $\hat{x}$, $\hat{p}$ and $\hat{H}$ don't commute together. If we consider the uncertainty for Hamiltonian and momentum, then we have $\Delta p→\infty$ but if we take uncertainty between momentum and position we have $\Delta p→0$. I would be grateful if help me what is wrong here?

 

[anuttarasammyak]

Momentum and coordinate are conjugate variables which satisfy the Kennard inequality. Hamiltonian and momentum, Hamiltonian and coordinate are not conjugate variable pairs.

 

[Lord Jestocost]

Momentum uncertainty isn't identical for position and Hamiltonian

Maybe, this can help. Check chapter “10.2 Compatibility” in the lecture “Commutators and Uncertainty Principle”.

Durham University: https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/uncertainty.html

 

[PeterDonis]

If we consider the uncertainty for Hamiltonian and momentum, then we have $\Delta p→\infty$ but if we take uncertainty between momentum and position we have $\Delta p→0$.

How are you obtaining these statements? Please show your work.