Feynman Lectures and Uncertainty Principle

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I read the Quantum Physics section of the online version of Feynman lectures http://feynmanlectures.caltech.edu/I_02.html#Ch2-S3 and I don't understand how he can deduce electron momentum from the Uncertainty Principle. I agree that the momentum is uncertain but how can he deduce that it is very large ?

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"If they were in the nucleus, we would know their position precisely, and the uncertainty principle would then require that they have a very large (but uncertain) momentum"
 

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  • #2
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Ask yourself: How can the expected value of the magnitude of the momentum be smaller than its uncertainty? Think of a set of numbers with large standard deviation. This set could have zero average. [itex]\langle x\rangle=0[/itex], but it canot have small average of magnitude [itex]\langle |x|\rangle >> 0[/itex].
 
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  • #3
Avodyne
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It would have been better if Feynman had said "the uncertainty principle would then make it very likely that they have a very large (but uncertain) momentum"
 
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I read the Quantum Physics section of the online version of Feynman lectures http://feynmanlectures.caltech.edu/I_02.html#Ch2-S3 and I don't understand how he can deduce electron momentum from the Uncertainty Principle. I agree that the momentum is uncertain but how can he deduce that it is very large ?

This is the relevant content:
"If they were in the nucleus, we would know their position precisely, and the uncertainty principle would then require that they have a very large (but uncertain) momentum"
What Mr. Fenyman, must have, meant was -- A very large spread in the probability distribution of Momentum.
 
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  • #5
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What Mr. Fenyman, must have, meant was -- A very large spread in the probability distribution of Momentum.
That is how I understand the uncertainty principle. But does that really imply that we know something about the probability of individual momentum values ? Does that really imply that a large momentum is more likely than a small momentum ?
 
  • #6
kith
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Picture a Gaussian distribution. The probability to obtain a momentum in a certain interval is the area under the curve. Even though the most probable value may be zero, the area for a small interval around zero is much smaller than the remaining area if the spread is large wrt to this intervall.
 
  • #7
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Sometimes I am getting confused about this too and then this is my line of reasoning that clears my mind: we would like to probe very small distances e.g. is the electron at position [itex]x[/itex] or at position [itex](x+ 10^{-9})m[/itex] or in other words our [itex]\Delta x[/itex] is of the order of [itex]nm[/itex]. But how do we do that - in particle physics what we measure actually is energy and momentum(we cannot take a ruler and measure the distance between particles). Now by the Heisenberg principle [itex]\Delta E[/itex] is very big, of the order of [itex]GeV [/itex]. Which energy [itex]E [/itex] is this [itex]\Delta E[/itex] uncertainty of - well the energy of our electron (the one we want to determine the position of). So if we try to measure this energy we will get numbers spread from [itex](E - \Delta E)[/itex] to[itex](E + \Delta E)[/itex]. Now what if [itex]E [/itex]is very small, like only [itex]eV [/itex] - we did not manage to measure nothing here. The only way to get meaningful results will be [itex]E[/itex] to be bigger that its own error (so at least of order [itex]GeV [/itex]), otherwise we did not measured anything. I hope that this is helping.
 

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