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

electron: ${\frac {5.77\times 10^{-9}}{\sqrt {9.109\times 10^{-31}}}}=6.04\times 10^{6}{\text{m}}$

proton: ${\frac {5.77\times 10^{-9}}{\sqrt {1.673\times 10^{-27}}}}=1.41\times 10^{5}{\text{m}}$

one-gram object: ${\frac {5.77\times 10^{-9}}{\sqrt {0.001}}}=1.82\times 10^{-7}{\text{m}}$

entire universe: ${\frac {5.77\times 10^{-9}}{\sqrt {m}}}{\text{ as m }}\rightarrow {\text{ large }}\Delta x\rightarrow {\text{ very small}}$

This indicates that an electron and a proton will not be very localized at all. Their initial locations at the start of the universe will indicate very little about their current location. A one-gram object will be much more localized than a proton or electron. An example of a one-gram object is a paper clip. If a paper clip's location is uncertain to $1.82\times 10^{-7}{\text{m}}$, we would have a very difficult time recognizing this uncertainty in location at all. The mass of the entire universe is not known (since the size of the entire universe is not known either). However, in comparison to a one-gram object, the mass of the universe is very very large. As m becomes very large the uncertainty in position becomes very very small. The uncertainty in the universe's position becomes negligible.