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Looking at an MIT note on propagation of errors, the error of a product $m\cdot v$ works out to the sum of the percentage errors, so we can report the range of values as $m \cdot v =m\cdot v ~(1 \pm ~(\frac{\delta m}{m} + \frac{\delta v}{v})),$ in which $\delta m$, for example, is the absolute uncertainty in mass.

Here is my question. In textbook problems on the uncertainty principle, we may be given $\Delta x$, uncertainty in position, given a mass $m,$ and asked to find uncertainty in velocity $\Delta v,$ on the premise that $\Delta p = m\cdot \Delta v,$ and of course $\Delta x \cdot \Delta p \geq \hbar /2.$

This idea is given explicitly in Harris, Modern Physics, 2d Ed., p. 49: $\Delta v = \frac{\Delta p}{m}.$

By the same logic we would I think have $\Delta m = \frac{\Delta p}{v}$.

Since the mass is much greater than its uncertainty, and the velocity greater than its uncertainty, I am not at all sure why we can use (in this case) $m$ as a proxy for $\delta m.$ It seems we would need to know the uncertainty in $m$, not $m.$

Can someone explain this?

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If you were trying to measure the mass of the particle in your experiment, then you are right that one would have $\Delta p = m \Delta v + v \Delta m$. However, if $m$ is the mass of an elementary particle (e.g. an electron or a proton), then it has been measured very precisely by others, and you would just look up the previously measured value. Because of the precision of these measurements, you will typically have $\frac{\Delta m}{m} \ll \frac{\Delta v}{v}$, and therefore the $\Delta m$ term can (usually) be safely neglected.

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    $\begingroup$ This does answer my question, thanks. Will accept in due course. $\endgroup$ Commented Oct 20, 2024 at 20:14
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The website in question addresses how to combine uncertainties in a student laboratory experiment. The errors that result from the uncertainty principle are so far below the errors induced by somewhat lousy student laboratory equipment and somewhat lousy (perhaps more than somewhat lousy) student measuring techniques that the uncertainty principle simply doesn't apply here.

In addition, that MIT note implicitly assumes that, for example, a student is estimating the mass, mass error, velocity, and velocity error separately. Also keep in mind that the note uses the "approximately equal" sign ($\approx$) throughout. Those who want to do a more thorough error analysis will inevitably use statistical techniques. The MIT note provides some rough rule-of-thumb techniques that are "good enough" for initial undergraduate students.

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    $\begingroup$ This is a useful reminder +1). All I needed was the reason that $\Delta m$ was discarded, but, yes, the method is a simplification. I don't even know if the uncertainty principle lends itself to undergraduate labs. I think Prof. Lewin's demonstration of the reciprocal spreading of momentum and position for light ( youtube.com/watch?v=MeK0DV329mU ) is convincing, and maybe quantifiable? $\endgroup$ Commented Oct 20, 2024 at 23:07

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