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The Heisenberg uncertainty principle applies here… you cannot make a measurement which is both precise in frequency and compact in time. This should be, in som

The Heisenberg uncertainty principle applies here… you cannot make a measurement which is both precise in frequency and compact in time. This should be, in some sense, natural to people who understand how a Fourier transform works… basically, when you apply a “window” function to measure a signal over a limited period of time, you end up with the signal smeared in the frequency domain.

pyuser583 5y ago
But I’m this case, it’s not compact in time. It’s talking about billions of years. So can it be precise in frequency? Am I misunderstanding?
- klodolph 5y ago
  "One second per 300 billion years" is really a statement about the bandwidth of the signal that the clock generates, translated for lay readers. We don't know what the actual bandwidth of the signal is, since it's not stated in the article. Bandwidth is measured in Hz. In order to measure the bandwidth of the signal (and how accurate the clock is), you have to measure the clock over a period of time. The shorter that period of time, the worse your measurement will be. The smaller the bandwidth, the more time you need to measure it. Note that a signal with 5 Hz bandwidth and 5 Hz center frequency would make a very terrible clock, while a signal with 5 Hz bandwidth and 5 GHz center frequency would make a very nice clock. The bandwidth is the same in both cases, and the amount of time required to measure bandwidth is the same, but they make very different clocks.
  - oshiar53-0 5y ago
    Does this mean that 0 Hz bandwidth leads to the ideal clock?
    - rcxdude 5y ago
      Yes, since that corresponds to a sine wave with fixed frequency extending forwards and backwards in time infinitely. (NB: it does depend on how you measure it. To truely get this you would also need to measure the clock output for all time).
  - pastage 5y ago
    What does center bandwidth mean here? My understanding of this from the domain of GPS time, has always been; you can have a good clock (GPS is slow, accurate over a long time) or a good ocillator (fast, inaccurate over along time). But together you can have both.
    - klodolph 5y ago
      Do you mean “center frequency?” A signal that goes from 9 Hz to 11 Hz has a center frequency of 10 Hz and a bandwidth of 2 Hz.
  - deleted 5y ago
    [deleted]
- myself248 5y ago
  The "over 300 billion years" kind of headlines always bother the crap out of me. The clock won't run for 300 billion years. It'll probably run for ten or twenty. But during that time there will be several copies of it built, and they'll run in several locations around the world, and the outputs of those copies will deviate less than some tinysecond per longtime. Attoseconds per month or whatever. And the useful timescale is more like, how much do they deviate over five seconds? Because they'll be used for something like astronomical ranging. Say you have two radio telescopes at distant points on the earth, both observing a distant stellar source simultaneously. The long baseline forms one side of a triangle, and the distances from the two observatories to the star form the other two sides. The more precisely you can determine those distances, the better you can resolve the position of the source. So you've got coherent receivers that can measure individual wavefronts in the signal, but how coherent are they, anyway? That depends on the clock they share. But they're on opposite sides of the planet, they don't actually share a clock... And the measurement period over which you can compensate for other confounding factors (probe and measure the atmospheric distortions in the way, for instance) might be very brief, so the clocks' stability over brief periods is salient. This is my lay understanding, so if someone more versed in the art wants to chime in, I'm all ears!
graycat 5y ago
THANKS!!! I keep trying to get a good understanding of at least the beginnings, i.e., foundations, of quantum mechanics and of the Heiserberg uncertainty principle, but I get frustrated at everything I can find on the Internet including the lectures of courses at MIT. So, for me, the physics is an obscure disaster. But Fourier theory!!!! That's FINE!!! I went carefully through the very careful treatment of Fourier series in the third edition of the W. Rudin Principles of Mathematical Analysis and also his very careful treatment of the Fourier transform in his Real and Complex Analysis, also distributions in his Functional Analysis. Also read and used the Blackman and Tukey The Measurement of Power Spectra, wrote some corresponding software that pleased some people in the US Navy, etc. Then I took a course at the level of the second Rudin book, and in the class when the prof got to the Fourier transform one of the students, with a physics background blurted out "That's the Heisenberg uncertainty principle!" So, now as I look at the MIT lectures, etc., I expect to see a good, solid, clear connection with the Fourier transform -- but so far I've found NOTHING. Yup, so, thanks for confirming that, yup, the Heisenberg uncertainty principle is not some weird and obscure feature of nature but, really, just an elementary result of any elementary but precise math derivation of the Fourier transform!!!! But, looks like I will have to make the connections with the basics of quantum mechanics myself. Yup, multiply in the time domain is the same as doing a convolution in the frequency domain -- that is, if take the Fourier transform of a function zero outside of a finite interval, then in effect have multiplied the function by a box, 1 in the interval and 0 otherwise, and then for the transform are doing a convolution, that is, a weighted sum, of the transform of the box which is just a version of sin(x)/x, that is, with a peak at the origin and falling off rapidly away from the origin. That is, the weighted sum of the convolution is a "smearing out". So, on to the connection with the physics ..., hopefully.
- klodolph 5y ago
  If you think of the Fourier transform as letting you express a function as a sum of sine waves, then in quantum mechanics, you could also choose to express a wavefunction as a sum of energy levels—or as a sum of momentums, or a sum of positions. https://en.wikipedia.org/wiki/Conjugate_variables https://en.wikipedia.org/wiki/Conjugate_variables

pyuser583 5y ago

But I’m this case, it’s not compact in time. It’s talking about billions of years. So can it be precise in frequency? Am I misunderstanding?

klodolph 5y ago
"One second per 300 billion years" is really a statement about the bandwidth of the signal that the clock generates, translated for lay readers. We don't know what the actual bandwidth of the signal is, since it's not stated in the article. Bandwidth is measured in Hz. In order to measure the bandwidth of the signal (and how accurate the clock is), you have to measure the clock over a period of time. The shorter that period of time, the worse your measurement will be. The smaller the bandwidth, the more time you need to measure it. Note that a signal with 5 Hz bandwidth and 5 Hz center frequency would make a very terrible clock, while a signal with 5 Hz bandwidth and 5 GHz center frequency would make a very nice clock. The bandwidth is the same in both cases, and the amount of time required to measure bandwidth is the same, but they make very different clocks.
- oshiar53-0 5y ago
  Does this mean that 0 Hz bandwidth leads to the ideal clock?
  - rcxdude 5y ago
    Yes, since that corresponds to a sine wave with fixed frequency extending forwards and backwards in time infinitely. (NB: it does depend on how you measure it. To truely get this you would also need to measure the clock output for all time).
- pastage 5y ago
  What does center bandwidth mean here? My understanding of this from the domain of GPS time, has always been; you can have a good clock (GPS is slow, accurate over a long time) or a good ocillator (fast, inaccurate over along time). But together you can have both.
  - klodolph 5y ago
    Do you mean “center frequency?” A signal that goes from 9 Hz to 11 Hz has a center frequency of 10 Hz and a bandwidth of 2 Hz.
- deleted 5y ago
  [deleted]
myself248 5y ago
The "over 300 billion years" kind of headlines always bother the crap out of me. The clock won't run for 300 billion years. It'll probably run for ten or twenty. But during that time there will be several copies of it built, and they'll run in several locations around the world, and the outputs of those copies will deviate less than some tinysecond per longtime. Attoseconds per month or whatever. And the useful timescale is more like, how much do they deviate over five seconds? Because they'll be used for something like astronomical ranging. Say you have two radio telescopes at distant points on the earth, both observing a distant stellar source simultaneously. The long baseline forms one side of a triangle, and the distances from the two observatories to the star form the other two sides. The more precisely you can determine those distances, the better you can resolve the position of the source. So you've got coherent receivers that can measure individual wavefronts in the signal, but how coherent are they, anyway? That depends on the clock they share. But they're on opposite sides of the planet, they don't actually share a clock... And the measurement period over which you can compensate for other confounding factors (probe and measure the atmospheric distortions in the way, for instance) might be very brief, so the clocks' stability over brief periods is salient. This is my lay understanding, so if someone more versed in the art wants to chime in, I'm all ears!

graycat 5y ago

THANKS!!! I keep trying to get a good understanding of at least the beginnings, i.e., foundations, of quantum mechanics and of the Heiserberg uncertainty principle, but I get frustrated at everything I can find on the Internet including the lectures of courses at MIT. So, for me, the physics is an obscure disaster. But Fourier theory!!!! That's FINE!!! I went carefully through the very careful treatment of Fourier series in the third edition of the W. Rudin Principles of Mathematical Analysis and also his very careful treatment of the Fourier transform in his Real and Complex Analysis, also distributions in his Functional Analysis. Also read and used the Blackman and Tukey The Measurement of Power Spectra, wrote some corresponding software that pleased some people in the US Navy, etc. Then I took a course at the level of the second Rudin book, and in the class when the prof got to the Fourier transform one of the students, with a physics background blurted out "That's the Heisenberg uncertainty principle!" So, now as I look at the MIT lectures, etc., I expect to see a good, solid, clear connection with the Fourier transform -- but so far I've found NOTHING. Yup, so, thanks for confirming that, yup, the Heisenberg uncertainty principle is not some weird and obscure feature of nature but, really, just an elementary result of any elementary but precise math derivation of the Fourier transform!!!! But, looks like I will have to make the connections with the basics of quantum mechanics myself. Yup, multiply in the time domain is the same as doing a convolution in the frequency domain -- that is, if take the Fourier transform of a function zero outside of a finite interval, then in effect have multiplied the function by a box, 1 in the interval and 0 otherwise, and then for the transform are doing a convolution, that is, a weighted sum, of the transform of the box which is just a version of sin(x)/x, that is, with a peak at the origin and falling off rapidly away from the origin. That is, the weighted sum of the convolution is a "smearing out". So, on to the connection with the physics ..., hopefully.

klodolph 5y ago
If you think of the Fourier transform as letting you express a function as a sum of sine waves, then in quantum mechanics, you could also choose to express a wavefunction as a sum of energy levels—or as a sum of momentums, or a sum of positions. https://en.wikipedia.org/wiki/Conjugate_variables https://en.wikipedia.org/wiki/Conjugate_variables