Phy5645/uncertainty relations problem1

This problem taken from Eugen Merzbacher's Quantum Mechanics 3rd edition: Exercise 2.7

Make an estimate of the lower bound for the distance $\Delta$x, within which an object of mass m can be localized for as long as the universe has existed ($\approx 10^{10}$ years). Compute and compare the values of this bound for an electron, a proton, a one-gram object, and the entire universe.

For nonrelativistic particles: $|t|\ll {\frac {m\hbar }{(\Delta p_{x})^{2}}}$, which can be rearranged to ${\frac {\Delta p_{x}}{m}}|t|\ll {\frac {\hbar }{\Delta p_{x}}}$.

Since ${\frac {\Delta p_{x}}{m}}=\Delta v_{x}$, and $\Delta p_{x}\Delta x\cong \hbar \rightarrow {\frac {\hbar }{\Delta p_{x}}}\cong \Delta x$, we can write:

$|t|\Delta \nu _{x}\cong \Delta x$.

Replacing $\Delta p_{x}$ with $\Delta p_{x}={\sqrt {\frac {m\hbar }{|t|}}}$, the uncertainty in position at time $t$ becomes:

$\Delta x\approx {\sqrt {\frac {\hbar |t|}{m}}}={\sqrt {\frac {(1.055\times 10^{-34}J\cdot s)(3.15\times 10^{17}s)}{m}}}={\frac {5.77\times 10^{-9}}{\sqrt {m}}}$.

This is an estimate of the lower bound for the distance within which an object of mass m can be localized for as long as the universe has existed.

We then have the following masses: $m_{e}=9.109\times 10^{-31}kg,m_{p}=1.673\times 10^{-27}kg,1g=0.001kg,{\text{mass of universe}}\rightarrow {\text{large}}$