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shreder
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if you know the position why can't you know the momentum
isnt p=mv and v=dx/dt its the only doubt i have so far please help
isnt p=mv and v=dx/dt its the only doubt i have so far please help
shreder said:if you know the position why can't you know the momentum
isnt p=mv and v=dx/dt its the only doubt i have so far please help
ZapperZ said:This is where you need to describe the SCENARIO in which you made the measurements. It makes a difference in QM, because QM is all about CONTEXT!
If I have just ONE particle, let's say I want to measure its position. What do I do? In QM this makes a boatload of difference on HOW you measure something. I can, for example,make it pass a very tiny slit, so that when it pass through it, I know that it had that position. Call this position x1, and the uncertainty in that position is Delta(x), corresponding to the width of the slit.
Now, there's nothing to prevent me from measuring, as accurately as I want, the momentum px AFTER it passes through the slit. How do I measure this? I can do that by
(i) putting a detector/screen after the slit;
(ii) measure the location that the particle hit the detector/screen;
(iii) measure how much it has moved transversely (i.e. in the x-direction) when it was moving from the slit to the screen)
(iv) calculate the x-component of the momentum since I know the speed that this particle moves and its mass;
(v) include the detector uncertainty that will give me the uncertainty in the momentum that I just measured. I will provisionally call this DDelta(px) - the double "D" is not a typo.
Now, I've just demonstrated that, in this case, I had just measured both position, and the corresponding momentum, to arbitrary accuracy limited by my instrument. In fact, I will argue that those two uncertainties (the Delta's) have no correlation with each other at all! I can increase the detector's accuracy, say by reducing the cross-talk between each of the detector channel, and thus, make the "spot" that I see on the detector or screen smaller, thus reducing the momentum uncertainty. This has ZERO effect on the width of the slit that I had the particle passed through.
So I've just showed you a way to measure position and momentum of a particle, and the HUP doesn't come into play... or does it?
What if I shoot a SECOND identical particle with the identical initial condition, at the identical slit? THIS is where it gets interesting!
After passing through the slit, do you think the particle will land on the detector at the same location? Classical mechanics will say that it should. After all, all the conditions are the same. But this is not true in quantum particles and in a quantum system.
Where the particle will land depends on how small the width of the slit is, i.e. how small Delta(x) is! The smaller Delta(x) is, the LARGER the spread in where the particle will hit the detector/screen. Remember, this also means that there will be a larger spread in the value of the momentum of the particle after it passes through the slit!
To know the nature of this spread, you have to send many more particles through the slit. It is ONLY when you have a statistically significant number will you see that there is a "central" value of the most probable momentum, and there is a spread (similar to the Standard Deviation in statistics) in the value of the momentum. Now this spread in momentum IS the "Delta(px) in the HUP!
So what does that tell you? You can't get the HUP from just ONE single measurement of x and px. If you look at the definition of the HUP, you'll see <x> and <p> and <x2> and <p2>. These are averages, which makes very little sense when you make just ONE measurement of each! So the uncertainties here is the SPREAD in the values that one would have upon REPEATED measurement of the identical system.
Secondly, it also tells you how well you can predict the outcome of the next measurement. The smaller the slit width, the less likely you will be able where the next particle will hit the screen. In other words, your uncertainty of the momentum will be larger with smaller slit!
This, in essence, is the HUP, and I had just described to you the single-slit diffraction.
https://www.physicsforums.com/blog.php?b=4364 [Broken]
Zz.
atyy said:In this method to measure the position and momentum of a single particle, it seems that you assume classical equations of motion of the particle from the slit to the screen. However, with Bohmian mechanics, the particle does not follow classical equations of motion. Why are the classical equations of motion privileged over the Bohmian ones?
ZapperZ said:Because it works!
Zz.
atyy said:But does your picture say that single particles have trajectories?
ZapperZ said:It has after it landed at the detector.
The same way that I can say that the particle that passed through the slit had that position at that slit BECAUSE it passed through it, the particle that made the "spot" at the detector had a definite trajectory from the slit to the detector AFTER it has been detected. It is why, from my avatar, the ARPES data can measure its momentum and energy.
Zz.
atyy said:It had a trajectory from the slit to the detector after it was detected, but not before it was detected?
atyy said:@ZapperZ, in this method of assigning a trajectory to a single particle, it seems that one is doing two successive measurements of transverse position. If the screen is very close to the first particle, because the particle is in a position eigenstate just after the slit, one would expect the transverse position on the screen to be very close to the first position, is that right?
curious bishal said:You know, position means where it is right now. Let's take an example of electron. There are many factors which accounts for the uncertainity:-
1) If you calculate the position of the electron, it escapes out from the present position to the next one while you are calculating and it always escapes.
2) For calculating the velocity, you have to take account a certain distance and time to measure the velocity and as you measure the velocity , electron slips more...
3) There is a fine delay coming the light waves from the electron to you so that there is always a remarkable error measuring the position and velocity
and there are others, too
shreder said:if you know the position why can't you know the momentum
ZapperZ said:Of course. At some point, if you are too close, you will not get any transverse resolution, and thus, you will not get good information about the momentum.
I forget if there is a near-field solution to this, i.e. when the screen distance is comparable to, say, the slit width or the wavelength. I think there is. When you are that close, the detector is doing nothing more than confirming the position of the particle, rather than a detector where you can extract the momentum.
atyy said:If the classical equations fail when the screen is near the slit, why is it justified to use the classical equations to assign the entire trajectory from the slit to the distant screen, since the trajectory must start near the slit? Shouldn't at least the initial part of the trajectory near the slit be described by non-classical equations?
ZapperZ said:Fail? Who said it failed?
Remember, in classical E&M, there is a such a thing as a near and far-field solutions! The near solutions often looks different than the far solution. Furthermore, at some point, if you make the slit too big, you will no longer see the diffraction pattern, and thus, no more spread in the transverse momentum! This is the same thing as making the screen too close to the slit!
Do you now claim that the HUP "failed"?
You seem to have ignore a very important point that I made in one of my responses - IT WORKS! We have been using it for ALL ARPES measurement, whereby the we can gather information about the electron's transverse momentum when it makes an image on our CCD screen. There is no better verification of the validity of anything than that!
Zz.
atyy said:You yourself said it fails in near field.
ZapperZ said:Of course. At some point, if you are too close, you will not get any transverse resolution, and thus, you will not get good information about the momentum.
ZapperZ said:No, I said that the information about the momentum may not have the same resolution as the far field!.
atyy said:If the slit and screen are close, and I register the position of the particle on the screen with high resolution, did the particle have a trajectory from the slit to the screen that is described by the classical equations of motion?
ZapperZ said:Yup.
Zz.
freshman123 said:I believe that there will be some certain new theories taking place of quantum mechanics.To be frank, I am more inclined to the idea of determination even though quantum mechanics is used as the main tool in my research.
atyy said:So if the screen and slit are near, and I have accurate positions of the particle at the slit and at the screen, then I can assign an accurate momentum to the particle along its entire trajectory using the classical equations, since the classical equations hold, right? Let's call this the single particle momentum.
However, we also know that when the screen is near the slit, the ensemble of single particle momenta obtained this way does not accurately reflect the momentum distribution we expect for an accurate momentum measurement of the wave function at the slit.
So would you say here we have accurate single particle momenta (since the classical equations hold, and we have two accurate position measurements), but the ensemble of these accurate single particle momenta are not accurate momentum distributions?
ZapperZ said:Think about this: what is the difference between a slit of width 1 micron, and a screen placed 1 micron away, versus a slit of width 1 meter, and a screen placed 1 meter away?
Can you get accurate momentum spread and momentum information out of that? Or did you just make Δx so big that there are no more momentum spread to be had?
Zz.
atyy said:I should be able to use them to get an accurate measurement of the single particle momentum.
DevilsAvocado said:But never simultaneously, right?
atyy said:Of course the momentum distribution when the screen is near the slit is not accurate. But the position measurements on the screen can be accurate whether or not the screen is near or far. Since there are two accurate positions - one at the slit and one at the screen - then if the classical equations hold near when the screen is near, I should be able to use them to get an accurate measurement of the single particle momentum.
atyy said:You are quoting me out of context.
ZapperZ said:No you can't, because if it is too close, the particle can come from ANY part of the slit! You now have an ambiguous origin on how to calculate your classical trajectory!
You have no way of assigning a unique momentum to that spot on the screen anymore!
DevilsAvocado said:What am I missing in this screen/momentum scenario...??
...
I don't get it... :uhh:
DevilsAvocado said:What am I missing in this screen/momentum scenario...??
Take two identical cars, traveling 100 km on a test track, ending in a rock wall. Car #1 makes the distance on exactly 60 min, and gets completely demolished when crashing into the rock wall; average speed 100 km/h.
Car #2 also makes the distance on exactly 60 min, and gets a tiny dent when bumping into the wall; average speed 100 km/h.
What happened?
Car #1 went 100 km/h from start to finish. Car #2 went 110 km/h for 54 min, hit the brakes and then went 10 km/h for 6 min.
So how on Earth can a smash into a screen tell us anything about simultaneous position and momentum?? (unless you check the 'recoil' of the screen)
I don't get it... :uhh:
ZapperZ said:OK, I didn't realize this is that difficult to understand, considering that this is almost first-year physics.
The uncertainty principle, also known as Heisenberg's uncertainty principle, is a fundamental principle in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle.
The uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. This means that there is always a level of uncertainty in the measurement of momentum due to the limitations of our ability to measure position.
The uncertainty principle has significant implications in physics, as it challenges our traditional understanding of the physical world. It shows that at the subatomic level, particles do not behave in a predictable manner and that there will always be a level of uncertainty in our measurements. This principle has also led to the development of new theories and concepts in quantum mechanics.
While the uncertainty principle may seem to only apply to the microscopic world, it has indirect effects on our daily lives. For example, it is the reason why we cannot accurately predict the exact path of a hurricane or the exact location of an electron in an atom. It also plays a role in the development of technologies such as MRI machines and transistors.
No, the uncertainty principle is a fundamental principle in quantum mechanics and cannot be overcome. However, there are ways to minimize the uncertainty in measurements, such as using advanced technology and techniques. Additionally, the uncertainty principle only applies at the subatomic level and does not have a significant impact on our macroscopic world.