The Uncertainty Principle

[souixsie]

Hi there. Can someone please explain this?

[GutZ]

The simplest I can think of...there is no certainty in being to measure momentum or position. Once you measure one, the other becomes influenced by it and changes state of being. Something to do with waves; and their nature of collapsing when disturbed.

 

Don't take my word as truth though. That's how I understand which could be wrong (and most likely is). I just post because it seems people who get that first response in their thread, especially if it's wrong answer will be hounded to be corrected. Trust me

Edited June 30, 2009 by GutZ

[bombus]

GutZ, you're not wrong there.

 

However, the Uncertainly Principle is that electrons (or any sub-atomic particles) cannot have a definite momentum AND position. This is sometimes attributed to being a problem of hidden variables that detectors are not accurate enough to detect, or because the detection device itself affects the result, but while this is true enough, it's actually an intrinsic quality of reality at the sub-atomic level. There is simply no such thing as an electron (or any sub-atomic particle) that possesses both a precise momentum and a precise position. Heisenburg stated 'We cannot know as a matter of principle, the present in all its details'.

 

The Uncertainly principle does not work backwards in time though. An electron can be shown to have had both precise momentum and position in the past - i.e., we can know as a matter of principle the past in all its details.'

Edited June 30, 2009 by swansont
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[SH3RL0CK]

Well, it is a difficult thing to explain. It took me about 5 seconds to locate a decent explaination here:

 

http://en.wikipedia.org/wiki/Uncertainty_principle

 

You could have done the same through Wiki, or Google, or etc. It does get frustrating when people expect someone else to look up the answers they want, rather than trying to do so themselves.

 

Is this a homework assignment? If so, please do not copy from Wiki as I am sure the instructor will figure this out.

[souixsie]

I have seen it on wikipedia, but didn't understand it. I still don't!

[SH3RL0CK]

I have seen it on wikipedia, but didn't understand it. I still don't!

 

I see. Think of it this way: You are trying to measure an electron's position and momentum. Let's define momentum as the direction and speed the electron is travelling.

 

These are obviously related to each other; the position of the electron changes in response to its direction and speed.

 

As your instruments get better and better at determining the exact position of the electron, you lose ability to measure its momentum. Once you know exactly where it is, you know longer know where it is going at all. This is because the process of determining its position has changed where how fast and in what direction it is going. From Wiki:

The uncertainty principle is often stated this way:

 

The measurement of position necessarily disturbs a particle's momentum, and vice versa

This makes the uncertainty principle a kind of observer effect.

 

This explanation is not incorrect, ...

This explaination is however, incomplete without going further into the math; as is described in the wiki article. As a starting point, does this incomplete explaination make sense to you?

[swansont]

The Uncertainly principle does not work backwards in time though. An electron can be shown to have had both precise momentum and position in the past - i.e., we can know as a matter of principle the past in all its details.'

 

Cite? i.e. support for this statement?

[proton]

Hi there. Can someone please explain this?

The Heisenberg Uncertainty Principle (HUP) relates the standard deviations of collections of data of a two conjugate variables taken from a series of experiments.

 

Example: suppose you have a wave function [math]\Psi[/math](x, y) which represents a particle restricted to move along the x-axis. Execute a large number of position measurements, each time starting with the same wave function. Let x_i represent the position recorded during the ith experiment. The probability of measuring the position of the particle in the range x to x+dx is [math]|\Psi|^2[/math]. The expectation value of finite number of x measurements is defined as

 

[math] <x> = \sum x_i P_i[/math]

 

where P_i is the probability of measuring x_i (in the range x to x+dx)

 

Also form the expectation of x²

 

[math] <x^2> = \sum x^2_i P_i[/math]

 

For a continuous variable such as x the sums become integrals. Now form the quantity [math]\Delta x[/math] defined as

 

[math] \Delta x = \sqrt(<x^2> - <x>^2)[/math]

 

[math]\Delta x[/math] is called the uncertainty in x. Now do the same thing for momentum. There is a momentum wave function which is the Fourier transform of [math]\Psi[/math](x, t). Call this [math]\Phi[/math](x, t). Use this similar to the above to get [math]\Delta p[/math]. The HUP states that

 

[math]\Delta x \Delta p \le \hbar/2[/math]

 

Simply put, [math]\Delta x[/math] is a quantity which gives a measure of the spread of possible position measurements as reflected in a data set that would be obtained from a large number of identical experiments of the measurement of x when the system is initially in the state [math]\Psi[/math](x, t). Similarly, [math]\Delta p[/math] is a quantity which gives a measure of the spread of possible momentum measurement as reflected in a data set that would be obtained from a large number of identical experiments of the measurement of p when the system is initially in the state [math]\Phi[/math](p, t). The HUP relates these two values by the above inequality.

 

Important - It can't be stressed enough that the value of the uncertainty is not determined by the measurement process and thus not determined by how the measurement is done. It is a value that is intrinsic to the initial state of the system. Its value is uniquely determined by the wave function of the system. Different wave functions give different uncertainties. No matter how good your measurement process is, or your instruments are, you cannot reduce the uncertainty in the measured value. And when you measure a variable it does not reflect the precision of the measurement

 

People almost always seem to have the wrong idea about what the HUP is all about and this wrong idea is a result of the wrong idea of what uncertainty is. If you keep in mind that the uncertainty in a quantity is the same thing as the standard deviation and is thus statistical in nature then you'll be on the right track. It’s misleading to say that the measurement in one value affects the uncertainty in the other value. The actual situation is that the wave function determines the uncertainty and if you change the wave function so as to decrease the uncertainty then you end up changing the Fourier transform of the wave function and that results in an increase in the uncertainty of the conjugate variable.

Edited July 1, 2009 by proton

[iNow]

Yeah, that was way simpler than the wiki which the OP didn't get.

[bombus]

Cite? i.e. support for this statement?

 

Eddington - 1939: The philosophy of physical science: Tarner lectures 1938 Cambridge University Press

 

W. Heisenberg: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. In: Zeitschrift für Physik. 43 1927, S. 172–198.

 

W. Heisenberg (1930), Physikalische Prinzipien der Quantentheorie (Leipzig: Hirzel). English translation The Physical Principles of Quantum Theory (Chicago: University of Chicago Press, 1930).

 

Gribbon, J (1984) In search of Scrodinger cat: London. Black Swan

 

This is patently obvious anyway.

[GutZ]

It’s misleading to say that the measurement in one value affects the uncertainty in the other value. The actual situation is that the wave function determines the uncertainty and if you change the wave function so as to decrease the uncertainty then you end up changing the Fourier transform of the wave function and that results in an increase in the uncertainty of the conjugate variable.

 

 

I knew I was wrong, that it right there but I have no idea how to simplify it. Proton your knowledge is vst but you need to tone it down alittle for us to fully grasp!

 

I agree though to make that make sense to a laymen you probably need to make your own course worth of material.

 

Maybe if someone dumb down that part it might help, and you can help me understand it as well....two birds with one stone....or particle...

[souixsie]

Proton, you lost me there I'm afraid.

 

Bombus, your explanation is the easiest to understand - but I find it hard to believe.

 

What are electrons then?

[Sisyphus]

Electrons, like all particles, can in some ways be mathematically described as waves. Simplistically, a wave can't have a single, precise location unless it is infinitely "bunched up," in which case its wavelength is indeterminate, and it can't have a precise momentum unless it has a precise orientation and wavelength, in which case there is nothing to pin it down to any particular location whatsoever. When particles interact with something, depending on the nature of the interaction, they can be forced to have a position and/or momentum to greater or less degrees, but due to the nature of the wave, increasing the precision of one makes the other necessarily indeterminate.

Edited July 1, 2009 by Sisyphus

[proton]

I knew I was wrong, that it right there but I have no idea how to simplify it. Proton your knowledge is vst but you need to tone it down alittle for us to fully grasp!

It's hard to know how to explain something to a generic person. It's also hard to explain the HUP to someone who doesn't know probability. If I had a better understanding of your background I could try to modify it.

 

Otherwise let me try it again - I will assume that you have a basic understanding of QM. By that I mean that there is a wave function associated with a particle. Let us denote this function as [math]\Psi(x,t)[/math] (assuming that the particle is restricted to move on the x-axis). The physical meaning of the wave function is that the square of the magnitude indicates where the particle is likely to be found when the position is measured. I.e. the probability is proportional to [math]|\Psi(x,t)|^2[/math]. To be precise - the probability [math]P(x, dx, t)[/math] of finding the particle about the point x in a region of width dx is [math]P_x(x, dx, t) = |\Psi(x,t)|^2dx[/math]. There is a similar relationship for the momentum which is called the momentum distribution function. Let that be labeled [math]\Phi(x,t)[/math]. Then Let that be labeled [math]|\Phi(x,t)|^2[/math] tells you what the probability of measuring the momentum to be p in an interval of width dp. I.e. [math]P_p(p, dp, t) = |\Phi(p,t)|^2dp[/math].

 

If the wave function is large at x and smaller elsewhere the particle said to be localized. The more localized the posotopm is the less localized the the momentum is. The relationship between the two is given by HUP relation I posted above.

 

How's that?

Edited July 1, 2009 by proton

[swansont]

Eddington - 1939: The philosophy of physical science: Tarner lectures 1938 Cambridge University Press

 

W. Heisenberg: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. In: Zeitschrift für Physik. 43 1927, S. 172–198.

 

W. Heisenberg (1930), Physikalische Prinzipien der Quantentheorie (Leipzig: Hirzel). English translation The Physical Principles of Quantum Theory (Chicago: University of Chicago Press, 1930).

 

Gribbon, J (1984) In search of Scrodinger cat: London. Black Swan

 

This is patently obvious anyway.

 

 

Cites of books really need page numbers.

 

And no, it's not patently obvious (to me) that you can recreate the exact trajectory of an electron where no measurement has been made.

[GutZ]

I would need a deeper understanding of both math and physics to really get the gist of what you are saying.

 

There is one thing you guys can clear up though is the wave/particlce duality thing

 

Is it really that a wave and particle are the same thing?

 

as in:

 

--wave/particle---> ]------------O-----------[

 

Kinda like a bead on a string?

Edited July 2, 2009 by GutZ

[Severian]

I will have a go at explaining it.

 

Everything in the universe is really a wave, and the HUP is really just coming out of the properties of waves. Nothing more mysterious than that.

 

For example, a "particle" is something which we think of as being very localised in position. It is in one place. But of course, our measurement of that position is never infinitely perfect, so it is really just a wave which is very very peaked it the point we think the particle is. This is the wave/particle duality thing.

 

Now we need to know a bit about the properties of waves. A wave with a single precise momentum is what we call a plain wave and looks like the traditional sine wave that you see all over the place. The wave has infinite extent, never dieing off to zero, and looks the same everywhere (apart from the up-down motion). So we can't say where this wave is at all - it is everywhere. This is one part of the HUP - if the wave has a well defined momentum, it is spread out over all space.

 

Let's go back to the particle, where we know its position very precisely and the 'wave' is a very peaked object around the particle's position (and dies off rapidly as we move away from that position. This peaked wave can be decomposed into a sum of lots and lots of plain waves, each of which has a well defined momentum. This is called a Fourier decomposition. Since each of these plain waves has a different momentum, we can't use any one to define the momentum of the particle. It doesn't have a definite momentum because it is made up of lots of different momentum plain waves. And the more peaked the distribution becomes, the larger the range of momentum we need in our plan waves. This is the other side of the HUP - if we know position of the wave very well, we have no handle on the momentum.

 

Notice that this isn't just a measurement problem. The peaked wave doesn't have a well defined momentum, and the plain wave doesn't have a well defined position.

 

To Gutz: Wave-particle duality is really just something we use to frighten children and undergraduates. Everything is a (quantum) wave, period. The only sense in which we have point-like particles, is that when we measure their position, we find that we get an error on that position which is only limited by our measurement. We teach undergrads that the wavefunction has collapsed into a position eigenstate, but this isn't quite correct, since it has collapsed into a superposition of position eigenstates, localized around a point, with an variance given by the error of our measurement. So even right after a position measurement, particles are still waves.

Edited July 2, 2009 by Severian

[proton]

Everything in the universe is really a wave, and the HUP is really just coming out of the properties of waves.

I wouldn’t phrase it like that myself. Particles cannot be said to “really” be a wave. That contradicts the wave-particle duality. The whole idea is that they have wave properties and particle properties. Do you recall how Feynman explained this? From his Lectures, V-III page 1-1

Newton thought that light was made up of particles, but then it was discovered that it behaves like a wave. Later, however, (in the beginning of the twentieth century), it was found that light indeed sometimes behaved like a particle. Historically, the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave. So it behaves like neither. Now we have given up. We say: “It is neither.”

For example, a "particle" is something which we think of as being very localised in position. It is in one place. But of course, our measurement of that position is never infinitely perfect, so it is really just a wave which is very very peaked it the point we think the particle is. This is the wave/particle duality thing.

Actually our measurement of its position can be made with arbitrary precision regardless of what the uncertainty in position is.

 

To Gutz: Wave-particle duality is really just something we use to frighten children and undergraduates. Everything is a (quantum) wave, period.

Where did you get such an idea?? Nothing could be further from the truth.

We teach undergrads that the wavefunction has collapsed into a position eigenstate, but this isn't quite correct, since it has collapsed into a superposition of position eigenstates, localized around a point, with an variance given by the error of our measurement. So even right after a position measurement, particles are still waves.

Not true.

Your first comment is correct. When a measurement is made the state falls into one if its eigenstates, not a superposition of them. I.e. when the position is measured the state is |x>. Since no measurement has infinite precision there delta function might actually be approximated by an extremely narrow step function. But that doesn’t mean that it’s a wave and not a particle. And as I stated above, the imprecision of a measurement has nothing to do with uncertainty. It's quite possible for a system to have a finite uncertainty and for a measurement to be made in which the result is determined with infinite precision. This happens when there is a discrete spectrum of eigenvalues.

 

Nothing personal, but I’m afraid that you have some misconceptions about the wave-particle duality.

Edited July 2, 2009 by proton

[Klaynos]

Wave - particle duality is about 'things' having properties of classical waves and classical particles.

 

A wave in quantum mechanics is something that can exhibit both of those classical properties, it's something else.

 

Hence why Severian says "Everything is a (quantum) wave, period." The quantum in brackets is important for this discussion.

[proton]

Hence why Severian says "Everything is a (quantum) wave, period." The quantum in brackets is important for this discussion.

That doesn't appear to be what Severian meant since he wrote

So even right after a position measurement, particles are still waves.

[Severian]

I wouldn’t phrase it like that myself. Particles cannot be said to “really” be a wave.

 

Why not? Can you give me a single property that is not (quantum) wave like? You can't because our theories are built on Quantum Field Theory. Notice the word 'field' here. A field is an object with a value for each point in the space, just like a wave. These quantum fields are indeed waves.

 

That contradicts the wave-particle duality. The whole idea is that they have wave properties and particle properties.

They have properties like particles because they can be very concentrated energy densities, and because they can carry fixed quantum numbers.

 

Actually our measurement of its position can be made with arbitrary precision regardless of what the uncertainty in position is.

It can be made with arbitrary precision, but not infinite precision. If you say you want a measurement better than x length units >0, then you can in principle (though often not in practice) build a tool to do it, but you will never make x zero.

 

Where did you get such an idea?? Nothing could be further from the truth.

Have you ever studied Quantum Field Theory?

 

Not true.

Your first comment is correct. When a measurement is made the state falls into one if its eigenstates, not a superposition of them. I.e. when the position is measured the state is |x>. Since no measurement has infinite precision there delta function might actually be approximated by an extremely narrow step function.

Isn't that (more or less) what I said? Or are you agreeing with that? Or are you claiming that the wavefunction then actually is a Dirac delta function (an ill defined and non-physical function)? If so, why do you need to approximate it with a step function? Since you can't measure more accurately, surely your objection would be untestable, and therefore unscientific.

 

But that doesn’t mean that it’s a wave and not a particle.

It means that it is spread out over space. Not usually a property I would identify with a point particle. In technical terms, in momentum space, it would be a superposition of position space eigenstates with different eigenvalues.

 

And as I stated above, the imprecision of a measurement has nothing to do with uncertainty.

Well, it has nothing to do with the Heisenberg Uncertainty Principle. By that I mean, even if you could make an infinitely precise position measurement, you would then have a Dirac delta function, which is still a superposition of (infinitely many) momentum space eigenstates. So infinite precision would still give a HUP. When I talked about imprecise measurements, I was trying to be helpful by explaining things in a way which didn't need implausible measurements and ill defined functions, but apologies if that confused you.

 

It's quite possible for a system to have a finite uncertainty and for a measurement to be made in which the result is determined with infinite precision. This happens when there is a discrete spectrum of eigenvalues.

No it isn't. Please provide evidence of an experiment which has provided a position measurement with infinite precision. I would be very interested to hear about it.

 

Even if it were, it would not invalidate my point for reasons I described above.

 

Nothing personal, but I’m afraid that you have some misconceptions about the wave-particle duality.

Please debate the science, not the credentials of the debater.

[proton]

Why not?

I explained above.

Please provide evidence of an experiment which has provided a position measurement with infinite precision. I would be very interested to hear about it.

Please reread what I wrote, i.e. determined with infinite precision. Consider a measurement of the spin of an electron. There can only be two possible values which are theoretically determined with no uncertainty.

Please debate the science, not the credentials of the debater.

Credentials???? Just because it's my opinion that you have a misconception, as reflected in your response, it has absolutely nothing to do with credentials.

[Severian]

I explained above.

 

You didn't answer my question. Can you give me a single property that is not (quantum) wave like?

 

Please reread what I wrote, i.e. determined with infinite precision. Consider a measurement of the spin of an electron. There can only be two possible values which are theoretically determined with no uncertainty.

 

We were not talking about spin. We were talking about position (since that is relevant to the concept of particles). And anyway, the precision of the measurement makes no difference to my argument.

 

Credentials???? Just because it's my opinion that you have a misconception, as reflected in your response, it has absolutely nothing to do with credentials.

 

Irrespective of terminology, I hope you agree that the statement "you have some misconceptions about the wave-particle duality" has no worth in debate.

[proton]

You didn't answer my question. Can you give me a single property that is not (quantum) wave like?

You used quantum to be synonymous with wave so your question has no meaning. I already gave you an example, i.e. spin. Spin is not wavelike. Anything which has a discrete spectrum is like that. When a measurment is made on a discrete system for which there is a uncertainty in the quantity being measured one can theoretically obtain precise values. Hence the difference between uncertainty in a state and the absolute precision of measured value.

Irrespective of terminology, I hope you agree that the statement "you have some misconceptions about the wave-particle duality" has no worth in debate.

Debate!? Nobody said anything about debate. I was merely saying that your responses represented what I consider to be misconceptions. If it will help you move on then I rephrase my comment as "That remark represents a misconception."

 

Back to physics - Theoretically the eigenkets of position are defined as

 

X|x> = x|x>

 

where X is the position operator and x is an eigenvalue of position. An arbitrary state can be expanded in terms of these eigenstates as

 

[math]|\alpha> = \int dx|x><x|\alpha>[/math]

 

A highly idealized experiment might be to place a very tiny detector that clicks when a particle is at x and nowhere else. After it clicks we say that the state is represented by |x>. I.e. we say that [math]|\alpha>[/math] jumps to |x>. However in practice one normally can only locate the particle to a narrow interval about x. So the state goes from [math]|\alpha>[/math] to [math]\int dx|x><x|\alpha>[/math] where this is integrated over the width of the detector. If the particle is located within this range the detector clicks. This is, by definition, a particle property. Your assertion about variance means that the width is non-zero. But your comments about it suggest that there is no particle phenomena here which is misleading. The "click" means that the particle is localized in the sense defined above. That's what the term particle means and what was meant by the Dirac function being represented in practice by a step function. Interpretation is a key point here. Point particle means that there is no theoretical limit to the smallness of the detector.

 

The reader can read more about this in Sakurai in the section Position Eigenkets and Position Measurements.

 

No it isn't. Please provide evidence of an experiment which has provided a position measurement with infinite precision.

I was speaking in general terms, not about position. If you read my comment more carefully you'd have noticed that since I wrote This happens when there is a discrete spectrum of eigenvalues. Obviosly position has a continous set of eigenvalues. Edited July 2, 2009 by proton

[Severian]

You used quantum to be synonymous with wave so your question has no meaning.

 

How is "quantum" synonymous with "wave"?

 

I already gave you an example, i.e. spin. Spin is not wavelike. Anything which has a discrete spectrum is like that.

 

That is like saying "Red is not strawberrylike". Waves certainly can have angular momentum, and there is no reason they can't have spin. In fact, my use of quantum was exactly (in part) for this reason (as I stated earlier) because the waves can carry discrete quantum numbers. I am not saying that all measurable quantities are continuous - I am saying that the position space wavefunction is continuous.

 

Debate? Who said anything about debate????

 

Isn't this a forum for scientific debate, where we are encouraged to maintain the standards of debate? Maybe the moderators can answer that one.

 

 

Back to physics ... [snip]

 

A highly idealized experiment might be to place a very tiny detector that clicks when a particle is at x and nowhere else. After it clicks we say that the state is represented by |x>. I.e. we say that [math]|\alpha>[/math] jumps to |x>. However in practice one normally can only locate the particle to a narrow interval about x. So the state goes from [math]|\alpha>[/math] to [math]\int dx|x><x|\alpha>[/math] where this is integrated over the width of the detector. If the particle is located within this range the detector clicks. This is, by definition, a particle property.

 

Not necessarily. It is indeed a property of a particle, but it is also a property of a concentrated wave. Have you ever stood in the sea and been pushed over when a wave hit you? Was that then really a particle?

 

The "click" means that the particle is localized in the sense defined above.

 

Look, I use the word particle all the time in my daily life. I don't run around refusing to discuss "particle physics" because really everything is a wave. But I do, in the back of my mind, understand that particle is a rather poor description of an electron (or whatever) and these things are really quantum fields, aka waves.

 

It is interesting that you quoted Feynman earlier, because great scientist though he was, he did rather leave us with a problem. Namely that his Feynman diagrams lead our students to believe that particles are somehow like billiard balls, and we have to go to great lengths to explain that really these things are waves/quantum fields.

 

I was speaking in general terms, not about position. If you read my comment more carefully you'd have noticed that since I wrote This happens when there is a discrete spectrum of eigenvalues. Obviosly position has a continous set of eigenvalues.

 

Feel free to backtrack all you like, but your statement was made in a paragraph where you said "when the position is measured".