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In the lecture, the professor starts by saying that we have a particle with some momentum, but that we can't know the momentum of this particle with full certainty. So we have a particle with uncertainty ## \Delta p ##.

Now, he takes the energy of this particle, ## E = \frac{p^2}{2m} ## and asks how we might find the change in energy (or the uncertainty in energy). We take the differential of ## E ## to get ## \Delta E = \frac{p}{m}\Delta p ##.

My question is this: in the energy uncertainty relationship, we have a ## \Delta p ##. But we also have a ## p ##. We have this momentum term. But what does that refer to? Because we started this conversation saying we can't know the momentum exactly. And we called ## \Delta E ## the uncertainty in energy.

So what does it mean to have the ## p ## term directly in our ## \Delta E ## expression?

Thanks for any insight.