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1. Uncertainty Principle (What Fundamentally is the Issue?)
Hi guys,

I am currently conducting research for a 5,000 word EPQ (extending project qualification) entitled "Does God Play Dice With the Universe?".

I plan to follow the progressive view of the scientific community over time concerning the doctrine of scientific determinism, i.e. from Laplace to the observer effect to the uncertainty principle etc.

I was hoping you could help me with the uncertainty principle. I understand that the minimum uncertainty in position times the minimum uncertainty in momentum equals Planck's constant over 4 Pi.

Also, I think I have a sketchy understanding of wave functions (don't they basically just map out a probability distribution for where the particle can be found? But then how is the momentum calculated?)

However I don't know/understand the fundamental reason why particles don't have a definite momentum and position. I thought perhaps that particle-wave duality meant that particles behave like waves and therefore to calculate a momentum you need the wave to move a bit, which creates uncertainty in the position? However, doesn't that still imply the particle has a definite position?

I'm kinda confused... Any help much appreciated!

Thanks,
Top Hats
2. Re: Uncertainty Principle (What Fundamentally is the Issue?)
(Original post by Top Hats)
Hi guys,

I am currently conducting research for a 5,000 word EPQ (extending project qualification) entitled "Does God Play Dice With the Universe?".

I plan to follow the progressive view of the scientific community over time concerning the doctrine of scientific determinism, i.e. from Laplace to the observer effect to the uncertainty principle etc.

I was hoping you could help me with the uncertainty principle. I understand that the minimum uncertainty in position times the minimum uncertainty in momentum equals Planck's constant over 4 Pi.

Also, I think I have a sketchy understanding of wave functions (don't they basically just map out a probability distribution for where the particle can be found? But then how is the momentum calculated?)

However I don't know/understand the fundamental reason why particles don't have a definite momentum and position. I thought perhaps that particle-wave duality meant that particles behave like waves and therefore to calculate a momentum you need the wave to move a bit, which creates uncertainty in the position? However, doesn't that still imply the particle has a definite position?

I'm kinda confused... Any help much appreciated!

Thanks,
Top Hats
the wave function is much more fundamental than that. It is the object the encodes all information about the state and need not make any reference to position. The equations of motion in QM (non relativistic anyway) are linear so the most general solution is the sum of all the individual solutions. This means that wave functions will be the weighted sum of these individual solutions (which we call the pure states) where the coefficient of a particular pure state is it's probability amplitude. Therefore the wavefunction can be expressed as the sum of the pure states of momentum or the pure states of position but it remains the same object.
These pure states are eigenstates of the respective operator (in QM, observables become operators acting wavefunctions) i.e. states that satisfy:
where Q hat is the operator.
To go between these two representations (position and momentum), the standard way is to take the fourier transform of one to get the other.
The uncertainty principle comes about because the more you restrict the position the more unrestricted it's fourier transform is. Therefore you can never have a particle in one place moving with an exact momentum.
3. Re: Uncertainty Principle (What Fundamentally is the Issue?)
(Original post by ben-smith)
the wave function is much more fundamental than that. It is the object the encodes all information about the state and need not make any reference to position. The equations of motion in QM (non relativistic anyway) are linear so the most general solution is the sum of all the individual solutions. This means that wave functions will be the weighted sum of these individual solutions (which we call the pure states) where the coefficient of a particular pure state is it's probability amplitude. Therefore the wavefunction can be expressed as the sum of the pure states of momentum or the pure states of position but it remains the same object.
These pure states are eigenstates of the respective operator (in QM, observables become operators acting wavefunctions) i.e. states that satisfy:
where Q hat is the operator.
To go between these two representations (position and momentum), the standard way is to take the fourier transform of one to get the other.
The uncertainty principle comes about because the more you restrict the position the more unrestricted it's fourier transform is. Therefore you can never have a particle in one place moving with an exact momentum.

However, I'm not entirely sure I understand...

I've just finished my first year of 'A' levels so I'm not too clued up on Fourier transformations and the like. Regarding wave functions, I may be getting them confused with a graph of probability against position that I saw in a book. In simple terms, what does a wave function look like, and what information does is show?

As for the uncertainty principle, I now understand mathematically about why both the position and momentum cannot be defined, but in real physical terms why is this? Visually, what's actually going on?

Thanks for your help, I'm a bit of a noob here!
4. Re: Uncertainty Principle (What Fundamentally is the Issue?)
The uncertainty principle applies in a sense to all objects, not just quantum scale things, if you are proper fussy about things.

Here's a scenario where we can determine the uncertainty principles relating of position and momentum, without having to be considering quantum weirdness:

We have a ball floating through space; and now we wish to find its momentum, and where it is. First, position. We wish to measure position at a time t, and we want to measure the position as quickly as possible, cause in the time taken to measure position, (say, dt), the particle will move v*dt (so we get some inherent uncertainty), so ideally, we would be like to measure in zero time. But then, if you have no time, then we can't know v, and hence can't know momentum (as P=mv).

On the other hand, for momentum, P=mv, so to find P we must have m (take that as a given) and v, but to know v we must measure distance travelled and time elapsed. The issue here is, though, if you want this value to be precise, then you must measure over infinite time. Anything less, and you will have some measure of uncertainty - you're ignoring some portion of it's existence! So if that's how you get precise momentum, how can you get position simultaneously? You can't at all. It will take on all possible positions in it path as you measure momentum! In the classical world, we have approximations that are by no means exact, and if you were to take out a quantum ruler and clock, will find them wildly varying around the true values (if you could know them).

When it comes to quantum, there is an added isse (and why it's so important to the theory); There is unavoidable uncertainty. Measure position in an instant or momentum over an infinite amount of time if you wish, but that relation is saying you still won't know it precisely. Another, and in my opinion much more interesting uncertainty principle, is for energy and time. Now that has some serious consequences!
Last edited by FireGarden; 01-08-2012 at 00:00.
5. Re: Uncertainty Principle (What Fundamentally is the Issue?)
(Original post by FireGarden)
The uncertainty principle applies in a sense to all objects, not just quantum scale things, if you are proper fussy about things.

Here's a scenario where we can determine the uncertainty principles relating of position and momentum, without having to be considering quantum weirdness:

We have a ball floating through space; and now we wish to find its momentum, and where it is. First, position. We wish to measure position at a time t, and we want to measure the position as quickly as possible, cause in the time taken to measure position, (say, dt), the particle will move v*dt (so we get some inherent uncertainty), so ideally, we would be like to measure in zero time. But then, if you have no time, then we can't know v, and hence can't know momentum (as P=mv).

On the other hand, for momentum, P=mv, so to find P we must have m (take that as a given) and v, but to know v we must measure distance travelled and time elapsed. The issue here is, though, if you want this value to be precise, then you must measure over infinite time. Anything less, and you will have some measure of uncertainty - you're ignoring some portion of it's existence! So if that's how you get precise momentum, how can you get position simultaneously? You can't at all. It will take on all possible positions in it path as you measure momentum! In the classical world, we have approximations that are by no means exact, and if you were to take out a quantum ruler and clock, will find them wildly varying around the true values (if you could know them).

When it comes to quantum, there is an added isse (and why it's so important to the theory); There is unavoidable uncertainty. Measure position in an instant or momentum over an infinite amount of time if you wish, but that relation is saying you still won't know it precisely. Another, and in my opinion much more interesting uncertainty principle, is for energy and time. Now that has some serious consequences!
Very good points
tbf the energy time inequality is to be expected if you consider that momentum is the generator of infinitesimal displacements in space and the hamiltonian is the generator of infinitesimal time displacements.
6. All I can suggest is make sure you understand the topic thoroughly before you start writing. If you're unclear or struggling to fill 5000 words, don't be afraid to change topics. Understanding is key, as it will also aid your presentation marks.

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7. Re: Uncertainty Principle (What Fundamentally is the Issue?)
(Original post by Top Hats)

However, I'm not entirely sure I understand...

I've just finished my first year of 'A' levels so I'm not too clued up on Fourier transformations and the like. Regarding wave functions, I may be getting them confused with a graph of probability against position that I saw in a book. In simple terms, what does a wave function look like, and what information does is show?

As for the uncertainty principle, I now understand mathematically about why both the position and momentum cannot be defined, but in real physical terms why is this? Visually, what's actually going on?

Thanks for your help, I'm a bit of a noob here!
Easy bit to clarify is that we need 3 coordinates for a point in space. The unusual bit for quantum mechanics is that the wavefunction maps out many electrons like the graph you've seen, but also the individual electron is associated with a wave. It boils down to what was discribed above about infinite time being required to get exact precision and therefore it is uncertain.

Are you saying you understand the mathematics, you're a cleverer man than me! In physics terms the quantum of energy, the particle e.g. the photon, is Planck's constant times frequency. If you're describing a particle with this constant then for the reasons mentioned above about time and measurement, you can't be exact like in classical physics.
8. Thanks for the help guys; I'm still not 100% confident but I understand a lot more than before which is down to your helpful posts.

Perhaps I'll post my final EPQ here when it's done so that future generations of googlers can nick all my hard work!

This was posted from The Student Room's Android App on my ALCATEL one touch 990