Heisenberg's Momentum-Position Uncertainty Principle

[Koshi]

I was reading about how Heisenberg found out that it is "impossible to determine simultaneously with unlimited precision the position and momentum of a particle" (Serway/Moss/Moyer, 174)My question is why is this true? I read that it had something to do with the large wavenumbers $$\Delta$$k, but I'm unsure exactly how that affects anything. I'm just a little hazy on the reason for why, even ignoring the error caused by measuring insturments, it would be impossible to measure two precise things at once.

 

[chrisk]

Consider an electron beam directed at a screen with a vertical slit of width delta y and a photographic plate a particular distance behind the screen. As the width of the slit narrows the accuracy of determining the vertical position of the electron increases. But, electrons behave like waves so a diffracton pattern will be created on the photographic plate. The narrower the slit, the wider the diffration pattern. This spreads out the vertical velocity distribution of the electron.

 

[karkas]

It is a property of Quantum Mechanical particles and/or systems (and is generalised in macroscopic phenomena too) that was mathematically derived by W.Heisenberg, and is experimentally accepted too.

It states that you can't measure either momentum or velocity with zero uncertainty without maximizing the uncertainty of either velocity or momentum, and vice-versa. This can become clear to you by this example:

Consider a particle and a physicist trying to measure it's position. The best way to directly measure it's position is by "sheding light" upon the particle. But even if you use only one photon to find out the particle's position, the photon will "move" the particle a tiny bit, giving it velocity. Therefore, you will then know something about the particle's position (it was "around" there a while ago) but you will not exactly know it's velocity.

I hope I've helped!

 

[Koshi]

But even if you use only one photon to find out the particle's position, the photon will "move" the particle a tiny bit, giving it velocity. Therefore, you will then know something about the particle's position (it was "around" there a while ago) but you will not exactly know it's velocity.

I hope I've helped!

As the width of the slit narrows the accuracy of determining the vertical position of the electron increases. But, electrons behave like waves so a diffracton pattern will be created on the photographic plate. The narrower the slit, the wider the diffration pattern..

Oh, that definitely clears things up. I was actually looking for good comparable examples like these. Thank you for the quick replies! I was really stuck on this.

 

[Dav333]

Reading this thread helped me understand more on the topic.

Easy to understand posts, thanks.

 

[karkas]

I don't believe that Recourse to Authority is the best way to explain things, but in this case an example of Heisenberg's Uncertainty principle is masterfully demonstrated by Richard Feynman in his book about Quantum Electrodynamics (QED) , pages 54-56.

Feynman uses his probabilistic arrow addition way throughout the book and the example.

 

[Gib Z]

Consider a particle and a physicist trying to measure it's position. The best way to directly measure it's position is by "sheding light" upon the particle. But even if you use only one photon to find out the particle's position, the photon will "move" the particle a tiny bit, giving it velocity. Therefore, you will then know something about the particle's position (it was "around" there a while ago) but you will not exactly know it's velocity.

That is not Heisenberg's Uncertainty Principle, its the Observer effect. Any proper explanation of Heisenberg's Principle will make clear use of the wave duality of particles, as this duality is the reason behind the effect.

 

[karkas]

The Heisenberg uncertainty principle is also frequently confused with the "observer effect". The uncertainty principle actually describes how precisely we may measure the position and momentum of a particle at the same time — if we increase the precision in measuring one quantity, we are forced to lose precision in measuring the other. Thus, the uncertainty principle deals with measurement, and not observation. The idea that the Uncertainty Principle is caused by disturbance (and hence by observation) is not considered to be valid by some, ^[who?] although it was extant in the early years of quantum mechanics, and is often repeated in popular treatments.

If that is indeed the case then Gib Z is correct. However (same source)

The uncertainty principle can be restated in terms of other measurement processes, which involves collapse of the wavefunction. When the position is initially localized by preparation, the wavefunction collapses to a narrow bump in an interval Δx > 0, and the momentum wavefunction becomes spread out. The particle's momentum is left uncertain by an amount inversely proportional to the accuracy of the position measurement:

$\sigma_p \,\ge\,\pi\hbar/\Delta x.$

If the initial preparation in Δx is understood as an observation or disturbance of the particles then this means that the uncertainty principle is related to the observer effect.

 

[Feldoh]

The uncertainty principle is MORE than a statement about measurement.

It says if we mess with the state as to cause a wave function collapse we can never precisely know momentum and position. There is intrinsic property of uncertainty in the wavefunction's collapsed state and has NOTHING to do with how accurate we can make a measuring stick.

 

[karkas]

I though the thread's purpose was to try and show how HUP can be understood. Relating it to the Observer's effect is a way to grasp the notion, in my opinion. Arguing over deep theoretical roots or implications of the Principle, considering my or any lack of Quantum Mechanics full understanding, could be called slightly irrelevant, although I might as well be wrong. :)

 

[Feldoh]

I though the thread's purpose was to try and show how HUP can be understood. Relating it to the Observer's effect is a way to grasp the notion, in my opinion. Arguing over deep theoretical roots or implications of the Principle, considering my or any lack of Quantum Mechanics full understanding, could be called slightly irrelevant, although I might as well be wrong. :)

I think it is relevant. The HUP is an intrinsic property of a microscopic system. Honestly asking anymore than that is the same as asking why F=ma works experimentally.