Heisenberg imcertainty principle (get it)

[mrfeathers]

I don't understand why it is so hard to find the exact position and velocity of orbiting electron. And also, why would we want to know it, if it is always moving? I am not trying to disprove it or anything, so don't make fun of me, i am an uneducated peon

 

[buddingscientist]

whats your idea of finding both the exact position and velocity of the electron?

how do you find the location of your keyboard? shine light on it (lamp or diffuse sunlight through the window) what happens if you try do the same for something tiny like an electron ?

 

[mrfeathers]

you should do that then, just prove the Heisenberg uncertainty therom but just looking at the orbiting electrons with your own eyes

 

[buddingscientist]

when you look at your keyboard, the photons of light that 'interact' with it (and ten hit our eyes and give us what we percieve as colour) are much much smaller than the keyboard.
but if we are trying to see an electron using a photon..
the wavelength ("size") of an photon is a lot larger than that of an electron, what implications does this have?

 

[Dr.Brain]

For complete understanding of Uncertainity Principle , check out :

http://www.doxlab.co.nr/

TOPIC#2 on the above site is Heisenberg's..

 

[ZapperZ]

For complete understanding of Uncertainity Principle , check out :

http://www.doxlab.co.nr/

TOPIC#2 on the above site is Heisenberg's..

I would not recommend the site listed above. It perpetuates the misconception of the HUP that I have written about[1], that it is the uncertainty in a single measurement. I have seen this mistake repeated several times within the past week on here.

The HUP is NOT about the uncertainty in INSTRUMENTATION or measurement. One can easily verify this by looking at HOW we measure certain quantities. It is silly for Heisenberg to know about technological advances in the future and how much more accurate we can measure things. This is NOT what the HUP is describing. The HUP is NOT describing how well we know about the quantities in a single measurement. I can make as precise of a measurement of the position and momentum of an electron as arbitrary as I want simultaneously, limited to the technology I have on hand. I can make improvements in my accuracy of one without affecting the accuracy of measurement of the other.

What the HUP is telling you is the difference between a classical system and a quantum system. In a classical system, if you have a set of identical initial condition, and you measure ONE observable, and then you measure another observable, you will continue to get the SAME value of that 2nd observable everytime you measure the same value of that 1st observable. The more accurate you measure the 1st observable, the more accuract you can predict the value of the 2nd observable the next time you want to do such a measurement. The only limitation to how accurate you can determine these observable is the limitation to your measuring instruments. But these limitations do NOT scale like the HUP. You don't make one worse as the other one becomes better, because these are technical issues and are not related to one another.

On the other hand, in a quantum system, under the IDENTICAL initial conditions, even if you measure a series of identical values for the 1st observable, the 2nd observable may NOT yield the identical result each time. In fact, as you narrow down the uncertainty of the 1st observable, the 2nd observable may start showing wildly different values as you do this REPEATEDLY. Therefore, unlike the classical system, your ability to know and predict what is going to be the outcome of the 2nd observable goes progressively WORSE as you improve your knowledge about the 1st observable!

Again, it has NOTHING to do with the uncertainty in a SINGLE measurement! It doesn't mean that if you measure with utmost accuracy the position of an electron, that that electron momentum is "spread out" all over the place. This is wrong! I can STILL make an accurate determination of that electron's momentum - only my instrument will limit my accuracy of determining that. However, my ability to know what its momentum is going to be the NEXT time I measure it under the idential situation is what is dictated by the HUP!

Zz

[1] [11-15-2004 09:26 AM] - Misconception of the Heisenberg Uncertainty Principle

 

[matness]

time is also an observable so why do we always take Δt= 0 while
Δx*Δp >= hbar/2 ?

 

[ZapperZ]

time is also an observable so why do we always take Δt= 0 while
Δx*Δp >= hbar/2 ?

Who is this "we"?

You will note that in typical school problems, you often work with a solution to the time-INDEPENDENT solution. It doesn't mean that the uncertainty in the time period is zero. It means that for that case, it is irrelevant since the description does not contain any time dynamics.

Zz.

 

[Nomy-the wanderer]

I've a simple answer, hope u won't consider it naive...

Most of the working theories right now weren't based on solid proofs but because we needed them, and some observations needed explanations...Theorists make theories, without confitmations, but we assume they r correct until they r proven wrong, technology gives us the chance to make sure that we r on the right track, we can't yet find the electron and the uncertainty principle is what really works right now...

See u when they find out something strong that would give us the opportunity to observe electrons..

 

[ZapperZ]

I've a simple answer, hope u won't consider it naive...

Most of the working theories right now weren't based on solid proofs but because we needed them, and some observations needed explanations...Theorists make theories, without confitmations, but we assume they r correct until they r proven wrong, technology gives us the chance to make sure that we r on the right track, we can't yet find the electron and the uncertainty principle is what really works right now...

See u when they find out something strong that would give us the opportunity to observe electrons..

What are "solid proofs"? Would the statement that says "IT WORKS" be considered as "solid proof"? How about if I point to you your modern electronics? Would that be considered as "solid proof"?

Most people forget that the HUP is a CONSEQUENCE, not the origin, of quantum mechanics! To find a problem in the HUP is to find a problem in QM. And unless people also forgot about the centenial year of QM in 1999, let me remind you that there was an almost universal acclaimed by physicists that QM is THE most successful theory SO FAR in the history of human civilization!

This means that if you think QM does not have "solid proofs", then other parts of physics suffer from even a worse level of lacking of solid proofs!

Please keep in mind that NO part of physics is considered to be accepted and valid until there are sufficient experimental/empirical agreement! In fact, many theories and ideas originally came out of unexpected experimental observation in the first place!

There is no lacking of "solid proofs" for QM, and the HUP. Why this is even brought up here, I have no idea.

Zz.

 

[matness]

"i" was thinking these as you said , until i see the word 'simultaneous' for measurements in the defn of HUP. if we say nearly simultaneous then i think there will be no problem(i hope so...)

Also there are different explanations for HUP, and maybe this the problem about understanding it. at first i was thinking i get it , but it didnt take a long time for me to confuse (because i am a beginner only)
Zz 's article is very helpful but if anyone can send a sketch of proof for HUP it will be more clear

thanks
n

 

[ZapperZ]

"i" was thinking these as you said , until i see the word 'simultaneous' for measurements in the defn of HUP. if we say nearly simultaneous then i think there will be no problem(i hope so...)

Also there are different explanations for HUP, and maybe this the problem about understanding it. at first i was thinking i get it , but it didnt take a long time for me to confuse (because i am a beginner only)
Zz 's article is very helpful but if anyone can send a sketch of proof for HUP it will be more clear

I don't understand. You want a "sketch of proof" for the HUP? What is this?

I think every student has either done, or seen the diffraction from a single slit. To me, this is a VERY clear example of the HUP! It just happens that we typically use wave description of light to account for such effects. But with the photon picture, the identical diffraction pattern can be directly obtained and the spreading is a direct consequence of the HUP!

More? The deBoer effect that is very pronounced in noble gasses is a direct consequence of the HUP. This leads to a correction to the internal energy (and thus, the specific heat capacity) of the gasses at very low temperatures. Only via taking into account such corrections can one obtain the experimental values!

But I think people pay waaaaay too much attention at disecting the consequences and forgetting the principles that CAUSE such consequences. The fact that this came out of "First Quantization" principle of QM that is based on [A,B] operations of two non-commuting observables is less understood by many who do not understand the formalism of QM. This is a crucial part of elementary QM with which a whole slew of consequences are built upon!

Zz.

 

[matness]

what i wonder is the mathematical part :

[A,B]= C --> ΔA * ΔB >= |<C>|/2

is it just a thm about standart deviation?

 

[seratend]

I don't understand why it is so hard to find the exact position and velocity of orbiting electron. And also, why would we want to know it, if it is always moving? I am not trying to disprove it or anything, so don't make fun of me, i am an uneducated peon

The second part answers to the first part of your post.

In QM, we may define the position of a particle. We may also define the momentum of a particle. However momentum is not the velocity of the particle. Many people in this forum always mix the momentum with the velocity (due to their equality for average values: i.e. Erhenfest Theorem) and tend to make incorrect deductions.
QM tells one thing: we cannot associate a classical path to a particle, hence we cannot define a couple (position, velocity) to the particle.
The HUP property applied to the (position, momentum) observables just highligh this fact: we cannot find a particle where both position and momentum have "defined" values equal to their mean values (i.e. through the Erhenfest Theorem, they have a classical path if is the case).

Seratend.

 

[seratend]

what i wonder is the mathematical part :

[A,B]= C --> ΔA * ΔB >= |<C>|/2

is it just a thm about standart deviation?

Yes.

Seratend.

 

[jackle]

I can make as precise of a measurement of the position and momentum of an electron as arbitrary as I want simultaneously, limited to the technology I have on hand.

What sort of technology do we have to achieve this?

 

[quetzalcoatl9]

The second part answers to the first part of your post.

In QM, we may define the position of a particle. We may also define the momentum of a particle. However momentum is not the velocity of the particle. Many people in this forum always mix the momentum with the velocity (due to their equality for average values: i.e. Erhenfest Theorem) and tend to make incorrect deductions.
QM tells one thing: we cannot associate a classical path to a particle, hence we cannot define a couple (position, velocity) to the particle.
The HUP property applied to the (position, momentum) observables just highligh this fact: we cannot find a particle where both position and momentum have "defined" values equal to their mean values (i.e. through the Erhenfest Theorem, they have a classical path if is the case).

Seratend.

my (rather naive) understanding of it is that in a classical system:

$$xp - px = 0$$

but accord to HUP:

$$xp - px \neq 0$$

so that measuring the position and then measuring the momentum, is not the same as measuring the momentum and then measuring the position. infact, they will always differ by $$\frac{ih}{2\pi}$$

position and momentum are not seen as real values, but as non-commutative operators. you can derive the schrodinger wave equation in a straightforward manner from this.

(this is one interpretation).

 

[Kruger]

so that measuring the position and then measuring the momentum, is not the same as measuring the momentum and then measuring the position. infact, they will always differ by

No, that's not correct. The HUP doesn't says something about only one simultaneoulsy measurment. It says something about a serie of simultaneously measurements (always the same conditions). You see?
If you make 1000 simultaneoulsy measurments (position and momentum) and you measure the position at each experiment exactly then you will get at each measurement of momentum a completely different value.

 

[Igor_S]

Many people in this forum always mix the momentum with the velocity (due to their equality for average values: i.e. Erhenfest Theorem) and tend to make incorrect deductions.

Umm, how can momentum be equal to velocity ? Ehrenfest theorem is about equality of average QM momentum, which is defined as $$-i \hbar \vec \nabla$$ and classical momentum $$m \vec v$$. Or generally it shows that average values of QM operators are equal to corresponding quantities in classical mechanics (I guess that's what you had in mind).

---
quetzalcoatl9,

HUP actually does say $$\left< (\Delta p_x)^2 \right> \left< (\Delta x)^2 \right> \neq 0$$. What you wrote, are commutation relations for operators, not a HUP (however, it's used in derivation of HUP). And you cannot get deltas by taking ONE particle. Let's say you have an instrument which determines position and momenutum up to 10 decimal places [in some units]. And let's say measurement of momentum of electron always gives one value, eg. 1.0000000001 [in some units]. Then you measure it's position 1st time and get let's say 0.0000000044 [some units]. A set of repeated measurement of position (with momentum set to 1.0000000001) will get you just random results, like 50.3243243212, 0, 13.1313131313, -400000.0000000001, etc. (but each of these measurements will have high precision, and that's the one that depends on the instrument itself!). However, calulating the average of the square of deltas of random numbers like that (you see, I HAVE to have more than one measurement to do average!), gives us a big number. This number appears in HUP.

When I let particle momentums have some distribution of let's say between 1.0 and 1.1 , the numbers I get for position will not be so random - they will have a peak value around some number. If I don't control momentum at all, but let particles pass a very small hole (that way I'm controling position) and measure momentum afterwards, I will get random results.

Hope this helps!

 

[seratend]

Umm, how can momentum be equal to velocity ? Ehrenfest theorem is about equality of average QM momentum, which is defined as $$-i \hbar \vec \nabla$$ and classical momentum $$m \vec v$$.

Ok, let's explain the "due to their equality for average values" in my previous post.
<P>=m.d<X>/dt= m<V> if no em field (case H=p^2/2m+V(q)).
=> implictly assuming m=1 units, we have <P>=<V> QED.

(I thought it was clear enough, but your post showed I was wrong with this assumption, now I hope it is clearer).

However, for a given relation V=dX/dt on operators (e.g. V=P/m), we have the eigenvalue relation v=dx/dt iff [V,X]=0 (if the operators are sufficently "gentle").

If V and X have not the same eigenbasis (in other words they do not commute: [V,X]=/=0) => the relation v=dx/dt is no more valid for the eigenvalues => we cannot associate a classical path to a particle.

HUP just reflects this fundamental property of operators.

Seratend.

 

[Igor_S]

Yes, it crossed my mind later that you assumed m=1, but in physics it's unusual to equal momentum and velocity (or it's done very rare, because mass is not fundamental constant).

I understand that in math, it's just a number, so who cares ?

 

[jackle]

I can make as precise of a measurement of the position and momentum of an electron as arbitrary as I want simultaneously, limited to the technology I have on hand.

What sort of technology do we have to achieve this?

I'd still really like to know the answer to this because it contradicts what I have been told by a trusted source. It is obviously a very fundamental principle of reality in our universe, so I think it is vital to know the facts. Ideally, I'd like a name of an established experiment where physicists can measure a complimentry pair simultaneously to an accuracy that demonstrates what you are saying.

Thanks.

 

[jackle]

Oh, by the way, you are also a trusted source, which is why I am being so persistent.

 

[ZapperZ]

I'd still really like to know the answer to this because it contradicts what I have been told by a trusted source. It is obviously a very fundamental principle of reality in our universe, so I think it is vital to know the facts. Ideally, I'd like a name of an established experiment where physicists can measure a complimentry pair simultaneously to an accuracy that demonstrates what you are saying.

Thanks.

But think about this (based on what I described in the single-slit experiment).

1. How well I know the location of a photon depends on how wide a slit I make, no? The smaller the slit, the more I know about where that photon was when it passed by it.

2. When it passed by the slit, what is its transverse momentum perpendicular to that slit? Unless its momentum changed between the slit and the detector, then I can make the assumption that this momentum remained the same between the moment it passed through the slit and the moment it hits the detector. All I need to do is figure out WHERE on the detector it hits. Then, using simple geometry, I know it's momentum in that direction. How well I determine that momentum depends on the resolution of my detector, i.e. how well can I determine where it hits the detector. The larger the number of pixels on my CCD camera, for example, the finer I can determine this location.

Both 1 and 2 allow me to determine the position and momentum of an INDIVIDUAL photon (or electron, or neutron) to arbitrary precision (i) independent of each other and (ii) depend entirely on the technology of the measurement apparatus. The HUP doesn't kick in here! The HUP kicks in on the subsequent photon IF I try to make a prediction on its momentum under the SAME slit size! The HUP also kicks in if I repeat this measurement many times till I get a statistical spread on the momentum value with a fixed slit width.

There are many state-of-the-art apparatus that use techniques. Photoemission spectroscopists are familiar with one - their hemisphrical detector, such as the Scienta SES electron analyzer, allows for the E vs k measurement simultaneously in one shot with electrons passing through a slit (see my avatar).

Zz.

 

[jackle]

I'll do some reading.

Thanks

 

[Antiphon]

ZapperZ-

I agree with what you say about the HUP having nothing to do with the accuracy of measuring devices.

However I strongly disagree with your interpretation of HUP. It very much
is about the impossibility of determining simultaneous position and momentum
in a single measurement.

You can easily determine both position and momentum with arbitrary
accuracy as long as you don't try to do it in a single measurement.
Here's a prescription for doing this.

1) Measure a particle's position with arbitrarily small uncertainty
2) Measure the same particle's position again with arbitrarily small uncertainty

The momentum the particle had between the two places is the
distance divided by the time.

You now know the position of the particle at two points with very
little uncertainty and you also know the momentum which the particle
had between (and arbitrarily close to) the two measurements. What
you cannot know is the momentum of the particle just after an accurate
position measurement.

It's the very act of the second measurement which renders the
momentum of the particle unkown just after that second measurement.

 

[ZapperZ]

You now know the position of the particle at two points with verylittle uncertainty and you also know the momentum which the particle
had between (and arbitrarily close to) the two measurements. What
you cannot know is the momentum of the particle just after an accurate
position measurement.

It's the very act of the second measurement which renders the
momentum of the particle unkown just after that second measurement.

Come again?

Unless I misread your description, you are implying that from the moment the particle passes through the slit and BEFORE it gets to the detector, the momentum of the particle is unknown or could CHANGE and is not what I measure at the detector. Is this true?

If this is true, then the momentum that I'm measuring CAN be dependent on where I put the detector from the slit. If I put it at 1 meter after the slit, that should give me a different momentum value than when I put it 12 meters after the slit. Last time I checked, this is not the case. Based on this, I can deduce that the momentum of the particle after it left the slit is the same no matter where I put the detector. Thus, I'm measuring the lateral momentum of that particle the instant it left the slit when its position confinment is applied.

Applying the HUP for ONE single measurement is absurd. I will remind you of the DEFINITION of the uncertainty of an observable in QM, which is:

$$(\Delta A)^2 = <A^2> - <A>^2$$

Now what are the averages of A and A^2 when you have just ONE measurement? And what is the uncertainty in THAT measurement?

Zz.

 

[Antiphon]

Unless I misread your description, you are implying that from the moment the particle passes through the slit and BEFORE it gets to the detector, the momentum of the particle is unknown or could CHANGE and is not what I measure at the detector. Is this true?

Here's what I'm saying. An accurate position determination by definition
peaks the wavefunction $$\Psi(x,t)$$ at point in space.

Mathematically this renders the momentum indefinite. There IS no one
unique value of momentum attributable to the particle after this
accurate position measurement.

In order to have a space-compressed wave function it will have a
momentum-space representation which is a superposition of a broad
spectrum of momenta. The actual (definite) momentum of the particle
will only come into existence at a future time IF a momentum measurement
is made.

This is the true meaning of the HUP.

 

[jackle]

I didn't get anywhere with my reading but I was able to find web references to Photoemission spectroscopists, hemisphrical detectors and Scienta SES electron analyzers.

I was a bit worried that if you took a calculation approach in general for obtaining momentum, you might get results that are never measured in practice. For example, if an electron tunnels through a barrier, it might seem to have traveled faster than light to get to the other side under some circumstances? Dunno.

 

[ZapperZ]

Here's what I'm saying. An accurate position determination by definition
peaks the wavefunction $$\Psi(x,t)$$ at point in space.

Mathematically this renders the momentum indefinite. There IS no one
unique value of momentum attributable to the particle after this
accurate position measurement.

In order to have a space-compressed wave function it will have a
momentum-space representation which is a superposition of a broad
spectrum of momenta. The actual (definite) momentum of the particle
will only come into existence at a future time IF a momentum measurement
is made.

This is the true meaning of the HUP.

But you have not addressed what I have pointed out. And what exactly do you mean by "definite"? Do you mean that in ONE shot, or do you mean my ABILITY to predict what the range of values would be IF I were to actually MAKE th measurement?

Take a free particle. Write down the wavefunction for that particle as a single plane wave. It can have a very DEFINITE momentum, or k (let's put a delta function there). However, if you try to PREDICT where it is located, you have a HUGE uncertainty. However, does this mean that I CANNOT measure the position at a given instant? There's nothing to prevent me from measuring a position of the particle as accurately as I want. Let's say I found it to be at x1 at time t1. If I prepare the IDENTICAL situation again, at the identical time t1, do you think I'll get x1 again? Classical mechanics says YES. Quantum mechanics says NO. This is because the situation has a very large uncertainty. The particle could be at x2 that is hugely different than x1. Your ability to know where the particle is is GONE because the momentum is so well-defined! THIS, is what is meant by the HUP. It isn't my ability to measure a single measurement.

Again, use the DEFINITION of the HUP. It is a series of measurements and one's ability to predict where the next one is going to be. It is not about ONE single measurement.

Zz.

 

[ZapperZ]

I didn't get anywhere with my reading but I was able to find web references to Photoemission spectroscopists, hemisphrical detectors and Scienta SES electron analyzers.

I was a bit worried that if you took a calculation approach in general for obtaining momentum, you might get results that are never measured in practice. For example, if an electron tunnels through a barrier, it might seem to have traveled faster than light to get to the other side under some circumstances? Dunno.

Er... you are forgetting that we use tunneling spectroscopy and photoemission spectroscopy to MEASURE the properties of various materials. If what we interpret out of those measurements are "never meausred in practice", then these materials that we are studying, when PUT into work in your electronics, would NEVER perform the way we have characterized them! In case you did not know, the EARLIEST confirmation of the correct band structure of semiconductors came from photoemission measurements! I don't think I need to explain further the importance of semiconductors AND the knowledge of their band structure, do I?

Zz.

 

[Antiphon]

But you have not addressed what I have pointed out.

I would if I could. But I'm only referring to one free particle and one measurement.
I'm not sure what your slits and such are all about.

And what exactly do you mean by "definite"? Do you mean that in ONE shot, or do you mean my ABILITY to predict what the range of values would be IF I were to actually MAKE th measurement?

By definite I mean that that outcome of the observation has a specific
and predictible numerical value.

I mean in one shot, if you localize a particle's position then you have obliterated
any hope of predicting a narrow range on the expecation value for the momentum operator.

You can always predict the range of values- and in this example the range on the
momentum is unlimited. It could be anything WHEN you finally get around to making
the measurment.

However, does this mean that I CANNOT measure the position at a given instant? There's nothing to prevent me from measuring a position of the particle as accurately as I want.

You almost have it Zz. You CAN measure it- you just can't PREDICT it. And as soon
as you make your very accurate position measurement you no longer have a definite
mometum- no more plane wave and no more definite momentum.

-All in a single measurement. THAT's the HUP.

 

[jackle]

...when PUT into work in your electronics, would NEVER perform the way we have characterized them...

I've often wondered why my computer keeps crashing in a very unpredictable way. I always put it down to software...

 

[ZapperZ]

I would if I could. But I'm only referring to one free particle and one measurement.
I'm not sure what your slits and such are all about.



By definite I mean that that outcome of the observation has a specific
and predictible numerical value.

I mean in one shot, if you localize a particle's position then you have obliterated
any hope of predicting a narrow range on the expecation value for the momentum operator.

You can always predict the range of values- and in this example the range on the
momentum is unlimited. It could be anything WHEN you finally get around to making
the measurment.



You almost have it Zz. You CAN measure it- you just can't PREDICT it. And as soon
as you make your very accurate position measurement you no longer have a definite
mometum- no more plane wave and no more definite momentum.

-All in a single measurement. THAT's the HUP.

If you have read my very early treatment of the single slit measurement, you would have noticed that I emphasized the ABILITY TO PREDICT several times!

Again, I have said repeatedly, that there is NOTHING to prevent you from making as accurate of a measurement of a SINGLE value of position and momentum. Period. I have said, again repeatedly, that one's ability is only limited by technology - how small a slit one can make, and how many fine pixels on the detector that the photon or electon hits! These are instrumental uncertainty, NOT the uncertainty in the HUP! [I feel as if I'm repeating this forever!]

But after one has made ONE measurement set (particle passing through slit, hits detector, so one has position and momentum), THEN, the very next one, if the identical sitation occurs and the particle passes through the slit, one's ability to PREDICT where the next one is going to hit the detector depends VERY MUCH on how small the slit is (i.e how small Delta(x) is!). If one does this a gazillion times, one will know Delta(x) by the size of the slit, but Delta(p) will be VERY large from the statistics alone if Delta(x) is small.

Again, nothing from the experiment above prevents me from obtaining a definite value of position and momentum from a single measurement. The uncertainty in these values are not governed by the HUP, nor are they related. If you are still claiming that they are, then tell me how my ability to change the width of the slit affects the density of the number of pixel on my CCD plate at the detector.

Zz.

 

[ZapperZ]

I've often wondered why my computer keeps crashing in a very unpredictable way. I always put it down to software...

Then if you think we have characterized it wrong, you should not fly commercially, drive your car, seek medical treatment, believe in the value of h and e, etc.

Zz.