4 ms·

fractional FT is the uncertainty principle then? you can know position locally, or momentum locally, or you can know them both blurrily with a rotated basis in

fractional FT is the uncertainty principle then? you can know position locally, or momentum locally, or you can know them both blurrily with a rotated basis in between

adonovan 3y ago
_Ordinary_ FT is the uncertainty principle. Fourier analysis of waves can tell you the frequency or the phase, but not both to arbitrary precision: the more precisely you can tell when the signal occurred (phase), the less precisely you can tell its pitch. (The spectrum of a sharp square wave is smeared across all the frequencies.) The QM wavefunction is a wave, and so the same applies. Phase is position and frequency is energy is velocity. So you can't know the position and the speed.
- ur-whale 3y ago
  > _Ordinary_ FT is the uncertainty principle. Yeah, I was about to jump in and say the same thing. More precisely, in Fourier theory (_regular_ Fourier Theory, not fractional), there is an inequality that can be proven independently of any physical interpretation and which directly implies the uncertainty principle as it's called in physics and QM. In other words, Heisenberg's uncertainty principle has basically nothing to do with physics or quantum mechanics, it's a basic property of the Fourier transform: As soon as two physical quantities are the FT of one another (the FT being almost an involution, i.e. the FT and the inverse FT are almost the same thing), they have to obey the uncertainty principle. Stated simply: the more localized a function (e.g. a small hump and almost zero everywhere else), the more spread-out its FT. And conversely: the more spread-out, regular and slow moving a function, the more localized its FT will be (all energy concentrated in a small region of the freq domain). Which, if you think about it for 5mn is quite intuitive: a function that is almost zero everywhere and suddenly exhibits a hump has to have a sudden rate of change. Which - very visually - implies high frequency components (high rate of change = high freq components).
  - wiml 3y ago
    The thing that quantum mechanics introduces here is the idea that there are conjugate variables (things which are the FT of each other), and that position and momentum is just one such pair.
    - ur-whale 3y ago
      > is the idea that there are conjugate variables (things which are the FT of each other) God forbid that physicists would use the existing lingo instead of inventing their own and create more confusion.

fractional FT is the uncertainty principle then? you can know position locally, or momentum locally, or you can know them both blurrily with a rotated basis in between

adonovan 3y ago

_Ordinary_ FT is the uncertainty principle. Fourier analysis of waves can tell you the frequency or the phase, but not both to arbitrary precision: the more precisely you can tell when the signal occurred (phase), the less precisely you can tell its pitch. (The spectrum of a sharp square wave is smeared across all the frequencies.) The QM wavefunction is a wave, and so the same applies. Phase is position and frequency is energy is velocity. So you can't know the position and the speed.

ur-whale 3y ago
> _Ordinary_ FT is the uncertainty principle. Yeah, I was about to jump in and say the same thing. More precisely, in Fourier theory (_regular_ Fourier Theory, not fractional), there is an inequality that can be proven independently of any physical interpretation and which directly implies the uncertainty principle as it's called in physics and QM. In other words, Heisenberg's uncertainty principle has basically nothing to do with physics or quantum mechanics, it's a basic property of the Fourier transform: As soon as two physical quantities are the FT of one another (the FT being almost an involution, i.e. the FT and the inverse FT are almost the same thing), they have to obey the uncertainty principle. Stated simply: the more localized a function (e.g. a small hump and almost zero everywhere else), the more spread-out its FT. And conversely: the more spread-out, regular and slow moving a function, the more localized its FT will be (all energy concentrated in a small region of the freq domain). Which, if you think about it for 5mn is quite intuitive: a function that is almost zero everywhere and suddenly exhibits a hump has to have a sudden rate of change. Which - very visually - implies high frequency components (high rate of change = high freq components).
- wiml 3y ago
  The thing that quantum mechanics introduces here is the idea that there are conjugate variables (things which are the FT of each other), and that position and momentum is just one such pair.
  - ur-whale 3y ago
    > is the idea that there are conjugate variables (things which are the FT of each other) God forbid that physicists would use the existing lingo instead of inventing their own and create more confusion.