Question

The position of a neutron in nucleus is known within an uncertainty of `5 xx 10^(-15) "m"`. At what speeds might we expect to find it moving?

Answer 1

`Deltav = 6.30 xx 10^6 "m/s"`

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Well, if the uncertainty in the position is known, we wouldn't know the actual speed, only the uncertainty.

`DeltaxDeltap_x >= ℏ//2`

is the Heisenberg Uncertainty Principle. We know that `p = mv` is the linear momentum, so its uncertainty is

`Deltap = mDeltav`

That means

`DeltaxmDeltav >= ℏ//2`

Or

`Deltav >= (ℏ//2)/(mDeltax)`

That means the minimum uncertainty in speed is:

`Deltav_"min" = ((6.626 xx 10^(-34) "J"cdot"s")/(4pi))/(1.675 xx 10^(-27) "kg" cdot 5 xx 10^(-15) "m")`

`= 6.30 xx 10^(6) "m/s"`

Or,

`color(blue)(Deltav >= 6.30 xx 10^(6) "m/s")`

But we know nothing about the actual value of `v`. All we can say is that the speed is:

`"speed" = (v pm 6.30 xx 10^6) "m/s"`