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*
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