re: The Heisenberg Uncertainty Principle
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re: The Heisenberg Uncertainty Principle
OK, so I just had an odd thought. As I understand it according to the Heisenberg Uncertainty Principle the more precisely we know the speed of a particle the less precisely we are able to know it's location and vice versa.
So what about a photon? We know it's speed with absolute certainty. it is c. So doesn't it follow that it should be impossible to know the position of it with any precision whatsoever?
What am I missing here?
So what about a photon? We know it's speed with absolute certainty. it is c. So doesn't it follow that it should be impossible to know the position of it with any precision whatsoever?
What am I missing here?
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Re: re: The Heisenberg Uncertainty Principle
I am not a physicist, but as I understand it, the more precisely we know the momentum of a particle the less precisely we are able to know it's location.
The momentum of a photon depends on its wavelength, which we do not know with absolute certainty.
The momentum of a photon depends on its wavelength, which we do not know with absolute certainty.
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Re: re: The Heisenberg Uncertainty Principle
Speed is a misnomer; an example of Heisenberg's Uncertainty Principle is that we cannot know both the position and momentum of a particle. While photons do not have rest mass, they do have momentum arising from their nonzero energy.
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Re: re: The Heisenberg Uncertainty Principle
curtis95112 wrote:The momentum of a photon depends on its wavelength, which we do not know with absolute certainty.
True, but unrelated. The wavelength of a photon should not be confused with the probability distribution of its quantum wave function.
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Re: re: The Heisenberg Uncertainty Principle
Unrelated how? Is a photon's momentum not h/lambda?
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Re: re: The Heisenberg Uncertainty Principle
gmalivuk wrote:Unrelated how? Is a photon's momentum not h/lambda?
It most certainly is, turns out.
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Re: re: The Heisenberg Uncertainty Principle
Drowsy Turtle wrote:True, but unrelated. The wavelength of a photon should not be confused with the probability distribution of its quantum wave function.
I haven't really studied quantum mechanics, but I thought the electromagnetic wave packet was related to the probability of detecting the photon at particular points at particular times. It's Fourier stuff, isn't it?
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Re: re: The Heisenberg Uncertainty Principle
FancyHat wrote:Drowsy Turtle wrote:True, but unrelated. The wavelength of a photon should not be confused with the probability distribution of its quantum wave function.
I haven't really studied quantum mechanics, but I thought the electromagnetic wave packet was related to the probability of detecting the photon at particular points at particular times. It's Fourier stuff, isn't it?
Pretty much, yes.
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Re: re: The Heisenberg Uncertainty Principle
Sure, this is all true, but it's not the same as quantum uncertainty is it? Not that I'm entirely sure how this works for photons anyway (although in cases like the double slit experiment, measurement collapses the waveform into specific locations along the wavefront, rather than into individual wave packets).
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Re: re: The Heisenberg Uncertainty Principle
The momentum of a photon depends on its wavelength, which you said was not relevant.
Re: re: The Heisenberg Uncertainty Principle
Drowsy Turtle wrote:Sure, this is all true, but it's not the same as quantum uncertainty is it? Not that I'm entirely sure how this works for photons anyway (although in cases like the double slit experiment, measurement collapses the waveform into specific locations along the wavefront, rather than into individual wave packets).
As I understand it, the electromagnetic wave packet corresponds to the quantum mechanical wave function.
A wave packet is equal to the sum of infinitely many waves, each of a single frequency and extending infinitely far back and forwards in time and space. In most places, these component waves destructively interfere, corresponding to zero probability of finding the photon, as a particle, in those places. But in the wave packet, the waves don't entirely destructively interfere, corresponding to nonzero probabilities of finding the photon there.
Each component wave has a corresponding momentum. Since these frequencies are spread over a range, there is a corresponding range of momenta for the wave packet. The shorter the wave packet in length, the broader the range of frequencies and momenta for the component waves. The shorter the wave packet, the less uncertainty there is for the position of the photon, but the greater the uncertainty for its momentum.
To have absolute certainty of the photon's momentum means only having one component frequency, but then you've got a wave that extends forever in both directions, and the photon could be detected anywhere along it. To have absolute certainty of the photon's position at a particular time means having an infinitesimally short wave function, which means having an infinitely wide range of component frequencies, which means the momentum could be anything.
Or something like that.
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Re: re: The Heisenberg Uncertainty Principle
^^ Makes sense...
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Re: re: The Heisenberg Uncertainty Principle
Also bear in mind that momentum is a vector, so to know the momentum of a photon (or any other particle) exactly we also need to know the direction it's traveling in.
Re: re: The Heisenberg Uncertainty Principle
The Heisenberg uncertainty relation is derived from non commuting operators acting on the wavefunction of a massive particle. A photon doesn't have mass in the traditional sense, or at least not in the context of the nonrelativistic Schrodinger equation. Thus, it is inappropriate in this context to apply the uncertainty relation because in Heisenberg's theory, fields are continuous even if states are quantized (that is, that light is a wave, not a particle).
Another way to think about this, is that in Heisenberg's theory our particles have position operators associated with them. In QED, there isn't really a position operator for a photon, which is the manifestation of the quantization of the EM field. This paper attempts to derive a new uncertainty relation for photons within this context: http://arxiv.org/abs/1110.2415.
On other matters, there are some other things stated in this thread that bothered me. Though momentum is a vector, when you calculate its uncertainty, you integrate over 3space to find the expectation values of <P^{2}> and <P>^{2} so the directions are averaged away. So direction doesn't directly affect the outcome of a momentum measurement.
Another way to think about this, is that in Heisenberg's theory our particles have position operators associated with them. In QED, there isn't really a position operator for a photon, which is the manifestation of the quantization of the EM field. This paper attempts to derive a new uncertainty relation for photons within this context: http://arxiv.org/abs/1110.2415.
On other matters, there are some other things stated in this thread that bothered me. Though momentum is a vector, when you calculate its uncertainty, you integrate over 3space to find the expectation values of <P^{2}> and <P>^{2} so the directions are averaged away. So direction doesn't directly affect the outcome of a momentum measurement.
Re: re: The Heisenberg Uncertainty Principle
soggybomb wrote:On other matters, there are some other things stated in this thread that bothered me. Though momentum is a vector, when you calculate its uncertainty, you integrate over 3space to find the expectation values of <P^{2}> and <P>^{2} so the directions are averaged away. So direction doesn't directly affect the outcome of a momentum measurement.
Ok. But still, a bunch of photons can be in a state where the magnitude of their momenta are fairly certain but their directions are not, eg in spherically symmetrical scattering. Or conversely, in a laser beam the uncertainty in the frequency of the photons (and hence their energy and the magnitude of their momentum) is fairly low, and if the beam is wellcollimated then the uncertainty in their direction is low, too.
Or have I screwed something up?
Re: re: The Heisenberg Uncertainty Principle
No not really, there are quantum limits to this but its just like how I can known to very high precision both momentum and position of macroscopic objects.
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