5 ms·

I'm reading through your description and this catched my eye: "In general, the smaller the wave, the faster it spreads out: this is the uncertainty principle."

by leni536 10y ago

I'm reading through your description and this catched my eye: "In general, the smaller the wave, the faster it spreads out: this is the uncertainty principle." This is not true, spreading out is due to dispersion. Why you don't see particle like wave packets on water surface? Also dispersion. In fact you can create particle like waves on a guitar string, which is pretty much dispersionless. Also you can describe massless particles in QM which are also dispersionless. The uncertanity principle is about the uncertanity of non-commuting observables at any given time (eg. position and momentum), it's not about the time dependence of the wave function.

deleted 10y ago
[deleted]
Steuard 10y ago
It's fairly common for people to relate this quantum phenomenon to the uncertainty principle, though. Someone asks: "Why can't I just say 'the particle is at rest in this small area' and expect it to stay there forever?" And one take on the answer is "Because you're imposing a very small uncertainty on the initial position, the initial momentum must have a very high uncertainty. That means there's a substantial probability that the particle will move away from its initial position over time, which corresponds to the wave function spreading out." You can certainly talk about it in terms of Fourier components or that sort of thing, too. There's a degree to which the uncertainty principle as applied to the position/momentum degrees of freedom is nothing but a statement about Fourier transforms. But I don't think it's a bad conceptual shorthand, especially for students who aren't yet experts in thinking about Fourier stuff.
- tamana 10y ago
  As a non Fourier expert, thinking about the general concept of the Fourier transform (spatial vs frequency coordinates),and how position and momentum are Fourier transforms of each other, was highly illuminating
sideshowb 10y ago
This certainly gives me food for thought. Are you saying then that by allowing people to directly observe the wavefunction I'm not really demonstrating uncertainty at all? As that only arises when you force observation via position/momentum? Also if a massless quantum particle is dispersionless does that mean you can say "the particle is in this small area and will stay here forever"? Doesn't that violate uncertainty?
- leni536 10y ago
  I would put it this way: Time dependence, therefore dispersion is in direct connection with the model Hamiltonian. The uncertainty principle is independent of time dependence and the model Hamiltonian. I think I made a mistake about massless quantum particles, since in 3D a packet disperses in all directions. But in 1D it works. It can't stay stationary, but it moves with "c" in either direction but it stays in a localized area, still the uncertainty principle still holds.
spott 10y ago
>it's not about the time dependence of the wave function. The uncertainty of position and momentum is fundamentally about the Fourier transform, and is definitely about the time dependence of the wave function. You don't even need a full description of quantum mechanics to get the uncertainty principle, "momentum is the Fourier transform of position" gets you there immediately. > This is not true, spreading out is due to dispersion. In this case, for a massive particle, the spreading out is due to the uncertainty principle. That uncertainty principle might be the cause of the dispersion, but you don't need to invoke anything else to get the spreading out. >Why you don't see particle like wave packets on water surface? Also dispersion. In fact you can create particle like waves on a guitar string, which is pretty much dispersionless. Also you can describe massless particles in QM which are also dispersionless. You can create particle like waves on water surface, on a guitar string (which IS dispersive, just not to first order), and in massless particles. They are called solitons, and are a pretty cool area of research.

by leni536 10y ago

deleted 10y ago

[deleted]

Steuard 10y ago

It's fairly common for people to relate this quantum phenomenon to the uncertainty principle, though. Someone asks: "Why can't I just say 'the particle is at rest in this small area' and expect it to stay there forever?" And one take on the answer is "Because you're imposing a very small uncertainty on the initial position, the initial momentum must have a very high uncertainty. That means there's a substantial probability that the particle will move away from its initial position over time, which corresponds to the wave function spreading out." You can certainly talk about it in terms of Fourier components or that sort of thing, too. There's a degree to which the uncertainty principle as applied to the position/momentum degrees of freedom is nothing but a statement about Fourier transforms. But I don't think it's a bad conceptual shorthand, especially for students who aren't yet experts in thinking about Fourier stuff.

tamana 10y ago
As a non Fourier expert, thinking about the general concept of the Fourier transform (spatial vs frequency coordinates),and how position and momentum are Fourier transforms of each other, was highly illuminating

sideshowb 10y ago

This certainly gives me food for thought. Are you saying then that by allowing people to directly observe the wavefunction I'm not really demonstrating uncertainty at all? As that only arises when you force observation via position/momentum? Also if a massless quantum particle is dispersionless does that mean you can say "the particle is in this small area and will stay here forever"? Doesn't that violate uncertainty?

leni536 10y ago
I would put it this way: Time dependence, therefore dispersion is in direct connection with the model Hamiltonian. The uncertainty principle is independent of time dependence and the model Hamiltonian. I think I made a mistake about massless quantum particles, since in 3D a packet disperses in all directions. But in 1D it works. It can't stay stationary, but it moves with "c" in either direction but it stays in a localized area, still the uncertainty principle still holds.

spott 10y ago

>it's not about the time dependence of the wave function. The uncertainty of position and momentum is fundamentally about the Fourier transform, and is definitely about the time dependence of the wave function. You don't even need a full description of quantum mechanics to get the uncertainty principle, "momentum is the Fourier transform of position" gets you there immediately. > This is not true, spreading out is due to dispersion. In this case, for a massive particle, the spreading out is due to the uncertainty principle. That uncertainty principle might be the cause of the dispersion, but you don't need to invoke anything else to get the spreading out. >Why you don't see particle like wave packets on water surface? Also dispersion. In fact you can create particle like waves on a guitar string, which is pretty much dispersionless. Also you can describe massless particles in QM which are also dispersionless. You can create particle like waves on water surface, on a guitar string (which IS dispersive, just not to first order), and in massless particles. They are called solitons, and are a pretty cool area of research.