Don't you need position to get momentum?

[gib65]

I'm thinking of the Heisenberg Uncertainty Principle here. I understand why an increase in the precision of position measurements results in a degradation of the precision of momentum measurements, but I don't quite understand how the reverse works - that is, getting precise measurements of momentum even though your position measurements are more uncertain. I mean, the only way I know of for getting momentum measurements is to get two position measurements, and using the amount of time between them, deriving the velocity. Multiplying that by the mass gives you momentum. But how are you supposed to get the two positions if the measurements you take of them are degraded? Is it a matter of taking many such measurements and deriving the average?

[D H]

You are thinking this way because you are extrapolating from our everyday world to the quantum world. In the everyday world, momentum is directly proportional to the time derivative of position. This is not the case in the quantum world. Here momentum is more a function of frequency.

 

A perfect position measurement is possible only if the wave packet that represents the particle collapses to a point in the spacial domain. This point particle is spread across the frequency domain. A perfect frequency (momentum) measurement is possible only if the wave packet that represents the particle collapses in the frequency domain. This single frequency particle is a perfect sinusoid in the spacial domain.

[pioneer]

One way to look at this is using a photography analogy. If I took a picture of an explosion, if my shutter speed was too slow, I would get motion blur. We don't exactly know the position of any fragment due to the blur. But based on the blur one can know we have momentum. If we use a shutter speed, that stops the motion, so we can know the exact position, the object looks like it is not moving.

 

Relative to the Heisenberg Uncertainty, the shutter speed problem is caused because the electron is moving about 1/14C, so it has a different observational reference or slight SR. If we measure position from our reference, we get motion blur. If we increase the shutter speed, we can't pin point the momentum. We as scientists remain in one reference, although the equipment can generate one of two shutter speeds. As such, we can never cause one photo to show exact position and momentum.

 

Let me give an analogy. Say we were in a train station and saw an SR train go by. It would appear distance contracted. From the point of view of the SR train, it would see the station look distance contracted. From our reference, the front of the train, with the train so contracted arrives just an instant before the middle. But from the train's reference, the front of the train is normal and arrives sooner that the center. This ambiguity is the uncertainty.

 

One possible analogy, is a fat person behind a holographic lense, that makes him look thin. I am not being rude calling him fat. He is our good humored lab assistant, risking life and limb, taking darts in the gut for science. If we shot a dart, at the side pants pocket, of the image of our assistant in the holographic lense, we would hit the lab assistant, somewhere in the gut. What we see, is not what our assistant sees. We are using dead aim for the pocket on the image, but our assistant thinks we are purposely trying to impale him.

 

Say we had a reference bell, that was a standard, and therefore in neither reference. We place it to the right of the holographic lense, but not in the view of the lense. If we asked out assistant to step to the right to trigger the sensor, the thin person in the lense, would need to move further. So the thin image would appear to move faster to the bell, or he would appear to get there after it was triggered, i.e., position or momentum.

[gib65]

One way to look at this is using a photography analogy. If I took a picture of an explosion, if my shutter speed was too slow, I would get motion blur. We don't exactly know the position of any fragment due to the blur. But based on the blur one can know we have momentum. If we use a shutter speed, that stops the motion, so we can know the exact position, the object looks like it is not moving.

 

This analogy I get - but is that the way it is in particle experiments? I mean, when you fire a long wavelength photon at an electron, say, attempting to get its momentum, does the photon come back bearing a "smeared out image" of the electron (the smearing representing the displacement of the electron)?

[swansont]

The HUP is inherent in the system, and really isn't about how you observe. The position and momentum operators don't commute, so there will always be an uncertainty. As D H has already said, there is a problem in trying to extrapolate from the classical world down to the quantum world.

[gib65]

The HUP is inherent in the system, and really isn't about how you observe. The position and momentum operators don't commute, so there will always be an uncertainty. As D H has already said, there is a problem in trying to extrapolate from the classical world down to the quantum world.

 

I understand this. But how does one do a momentum reading? I understand that one fires a long wavelength photon at the particle to be measured, and I assume that it returns (with degraded position reading, of course). How does the experimenter deduce the particle's momentum from this photon?

[CPL.Luke]

I'd imagine that its done by looking at the KE, as momentum is equal to (2m (KE))^1/2

[gib65]

Well, since no one seems to be able to answer my question, I'll venture a guess. I'm going to say that, concerning our current technology, momentum of a particle can't be measured. The HUP is true, first and foremost, mathematically - that is, it can be shown mathematically that as one increases the precision with which one measures position, the precision with which one measures momentum decreases in proportion - and visa-versa. This tells us that, if we were to fire long wavelength photons at it (thereby degrading our position readings), the particle to be measured can gain in momentum in virtue of collapsing in the frequency domain (as D H put it). But this effect is only inferred by the math. As for verifying this collapsed frequency observationally, we don't have the means to do so technologically. But the math tells us that it's there, waiting to be measured, and if the day ever came where we would be able to measure a particle's momentum, we would find that it can only be done at the cost of precise position readings.

 

I would really like for someone to challenge me on this.

[swansont]

Scattering a photon off of a particle can tell you about the position and the momentum. But when you maximize the precision of one measurement, you maximize the uncertainty of the other.

[gib65]

Scattering a photon off of a particle can tell you about the position and the momentum. But when you maximize the precision of one measurement, you maximize the uncertainty of the other.

 

Okay, but how? How does scattering a photon off a particle give you a measure of its momentum?

[swansont]

Okay, but how? How does scattering a photon off a particle give you a measure of its momentum?

 

Because you will shift the wavelength of the photon and it will scatter at some angle. Since energy and momentum will be conserved, this can give you the initial momentum of the electron.

[gib65]

Because you will shift the wavelength of the photon and it will scatter at some angle. Since energy and momentum will be conserved, this can give you the initial momentum of the electron.

 

Ah, thank you swansont. I finally feel enlightened . The shift in wavelength is the crucial piece of information carried by the photon that experimenters use to deduce momentum. That's exactly what I was looking for. thanks.