On 5/20/2025 4:35 PM, Alan Grayson wrote:
In general when noise is small it can be treated as additive so when you measure you get some eigenvalue+noise, not the true value. Of course if the system is in a single definite eigenstate, not a superposition of many eigenstates, you can repeat the measurement many times and the noise term will average to zero.On Tuesday, May 20, 2025 at 11:54:39 AM UTC-6 Brent Meeker wrote: On 5/20/2025 4:16 AM, Alan Grayson wrote:*Your attachment shows how to establish the HUP, not why there is a spread in momentum. Classically, energy and momentum are related by a simple formula. So if one wants to /prepare/ a system in some specific momentum, one needs to control the energy of the particle. Presumably, this can never be done precisely; hence we get the spread. Is this not a sufficient explanation for the spread? AG**As far as the HUP is concerned the cause of spread in momentum is that the spread in conjugate position must be finite, and vice versa. * *Are all the momenta in the spread, eigenvalues of the momentum operato*r*? AG**Yes. But they have different probabilities of being found when measured. Brent* *But if one always gets a spread, how can any particular momentum in the spread be measured? AG **You can't choose which value you get measuring a random variable. You just measure momentum and you get a certain value. Then you repeat the experiment and you get a different value. You repeat this a thousand times and you can plot the distribution function of momenta and measure the spread. Brent* * * *Presumably, if it's momentum that's being measured, and one always measures eigenvalues, why is the spread larger on "imprecise" measuring devices, as opposed to undefined "ideal" measurements? And what is an ideal measurement? AG **An ideal measurement is one that leaves the system in the eigenstate corresponding to the measured eigenvalue. It's effectively a preparation. So it excludes destructive measurement, like hitting photographic film. I was assuming ideal measurements. Of course in real measurements the instrument noise may be bigger than the interval between eignvalues and so introduces additional spread. Brent* * **But regardless of the increased spread, won't the noise still result in eigenvalues of the momentum operator? AG*
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