Estimating with uncertainty principle

In summary, the concept being discussed is that in order to have a well-defined position and momentum, the radius must be larger than the spread in position and the momentum must be larger than the spread in momentum. The uncertainty of momentum is used as a substitute for the actual momentum, which is explained by the fact that the average momentum in a closed orbit is 0, making the time-average of the momentum squared equal to the variance in the expectation value of the momentum.
  • #1
dsdsuster
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http://quantummechanics.ucsd.edu/ph130a/130_notes/node98.html

Can someone help me understand what's going on here?

He says "The idea is that the radius must be larger than the spread in position, and the momentum must be larger than the spread in momentum." which I suppose must be true in order to have a well-defined position and momentum? And then he uses the uncertainty of the momentum as a substitute for the actual momentum. This part in particular I don't really understand.

Thanks!
 
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  • #2
I think it's because the average momentum is 0 (Closed orbit), so the time-average of the momentum squared (Which is what actually enters the equation), is exactly the variance in the expectation value of the momentum.
 

1. What is the uncertainty principle?

The uncertainty principle is a fundamental principle in quantum mechanics that states that the more precisely the position of a particle is known, the less precisely its momentum can be known, and vice versa. This means that there is a limit to how precisely we can measure certain properties of a particle at the same time.

2. How does the uncertainty principle relate to estimating?

The uncertainty principle is relevant to estimating because it sets a limit on how accurately we can estimate certain properties of particles. For example, if we want to estimate the position of a particle, there will always be some uncertainty in its momentum. This means that our estimate will never be completely accurate, and we must take this uncertainty into account when interpreting our results.

3. What are the implications of the uncertainty principle for scientific measurements?

The uncertainty principle has significant implications for scientific measurements. It means that there will always be some degree of uncertainty in our measurements, and this uncertainty cannot be eliminated. This requires scientists to carefully consider and account for the uncertainty in their measurements when making conclusions and predictions based on their data.

4. How does the uncertainty principle impact our understanding of the physical world?

The uncertainty principle challenges our traditional understanding of the physical world. It shows that at a fundamental level, there are inherent limits to our knowledge and measurement of the universe. This principle has led to the development of new theories and models in physics, and has expanded our understanding of the nature of reality.

5. Can the uncertainty principle be tested and verified?

Yes, the uncertainty principle has been extensively tested and verified through experiments in quantum mechanics. These experiments have consistently shown that there is a fundamental limit to how precisely we can measure certain properties of particles. However, there are ongoing efforts to refine and improve our understanding of the uncertainty principle and its implications.

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