Phy5645/uncertainty relations problem1

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For nonrelativistic particles, , which can be rearranged to .

Since , and , we can write:

.

Replacing with , the uncertainty in position at time becomes:

.

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:

electron:

proton:

one-gram object:

entire universe:

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 then 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 large. As becomes very large the uncertainty in position becomes very small. The uncertainty in the universe's position becomes negligible.

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