Understanding the Measurement and Interaction of Electrons: A Beginner's Guide

[durant35]

Hi guys, I'm a newbie in quantum physics and it has blown my mind so far. I feel a bit confused because it seems very unintuitive, but I'm ready to learn more and I need some help on this forum. I have few questions.

1) I red that the wavefunction of an electron is spread all over the universe, so there's a probability that we can find the electron basically everywhere. How do physicists do the measurement to determine the position of an electron since on the common sense view it seems that they should eliminate every portion of space without the electron to find it? Any normal description?

2) How do we know that electrons interact with other stuff in everyday life (like air) since they can be anywhere so we can't specify the place of the interaction and by that the interaction itself?

3) I also red that the interaction between the electron and other things causes the electron to decohere, and serves as a measurement, so how does this 'real life' measurement differ from the measurement in the lab?

4) Does the electron probability cloud reduce when in interaction with other stuff, so for instance can I say that an electron from the atom in my desk can now be in, let's say, South Africa? It seems pretty confusing.

5)What about the nucleus? In the modern atomic model it is said that the electron doesn't have a definite position, but the nucleus is regarded as almost fixed in position and momentum. What is the uncertainty in position of the nucleus and how can we know where the atom as a whole is?

Again, I know that this questions may seem weird but I'm a beginner so thanks in advance for the patience, I hope that some explanation will clear up the mess in my head.

 

[BvU]

Hi Durant,

And an extra welcome to this weird world of QM. All that counts there is amplitude of wave function and probability. And that's exactly what will take care of your mental sanity: the probability of finding an atom from your desk on the south pole is rather low. (understatement of the century). Let alone the probability of finding the entire desk there. You could look out for a few exercises on the Broglie wavelength to reassure you.

I had a lot of benefit from these Feynman lectures (i'll find the person from whom I got the link and will credit him/her) [edit] it was Simon Bridge -- Richard Feynman talks about photons, but later on you understand that this is true for all and everything (except radioactivity and gravity ).

 

[mfb]

I red that the wavefunction of an electron is spread all over the universe, so there's a probability that we can find the electron basically everywhere.

This would be true for an isolated atom (or a free electron) if you wait for infinite time. In practice, no particle is completely isolated, and we cannot wait for an infinite time. So most particles are quite localized. We cannot know their exact position, but we can have a good idea where they are approximately. And we can measure it: a measurement leads to a specific position - not by eliminating all other options, but by simply interacting with the electron in the right way, e.g. with x-rays.

so how does this 'real life' measurement differ from the measurement in the lab?

There is no difference.

4) Does the electron probability cloud reduce when in interaction with other stuff, so for instance can I say that an electron from the atom in my desk can now be in, let's say, South Africa? It seems pretty confusing.

Let's take a hydrogen atom in the ground state and assume it is the only atom in the world. Then you have something like 90% probability that (if you measure the position) the electron will be closer than 0.1 nanometer to the nucleus. The probability that it is closer than 1 nm is already something like 99.99%. Closer than 10 nm? 99.9999999999999% or probably even more. I made up those numbers, but you get the idea. The probability that the electron is a macroscopic distance away is completely negligible.

5)What about the nucleus? In the modern atomic model it is said that the electron doesn't have a definite position, but the nucleus is regarded as almost fixed in position and momentum. What is the uncertainty in position of the nucleus and how can we know where the atom as a whole is?

The same things apply to the nucleus, but as it is more massive, its localization in space is even better.

 

[durant35]

What about macroscopic objects? I've red that interactions between particles and the fact that there are so many of them elimininate uncertainty in position so that the macro object has an uncertainity in position in the range of 10^-30mm. What does that exactly mean, that we can find the object only in the range of its size plus the deviation I mentioned?

 

[mfb]

If macroscopic objects would be elementary particles with the same mass, that distance would be a reasonable possible uncertainty. Actual macroscopic objects are not elementary particles, they have a finite size and their individual components do not have fixed distances to each other. Talking about the position of a desk with a precision of 10^-30 mm just does not make sense.

 

[durant35]

"
This is perhaps the most famous equation next to E=mc2 in physics. It basically says that the combination of the error in position times the error in momentum must always be greater than Planck's constant. So, you can measure the position of an electron to some accuracy, but then its momentum will be inside a very large range of values. Likewise, you can measure the momentum precisely, but then its position is unknown.

Notice that this is not the measurement problem in another form, the combination of position, energy (momentum) and time are actually undefined for a quantum particle until a measurement is made (then the wave function collapses).

Also notice that the uncertainty principle is unimportant to macroscopic objects since Planck's constant, h, is so small (10-34). For example, the uncertainty in position of a thrown baseball is 10-30 millimeters."

This is the quote that I was referring to. Can you analyze it and compare it with your answer? What are the uncertainities in position and momentum of macroscopic objects then?

 

[vanhees71]

Hi guys, I'm a newbie in quantum physics and it has blown my mind so far. I feel a bit confused because it seems very unintuitive, but I'm ready to learn more and I need some help on this forum. I have few questions.

Don't worry, it's normal to find QT strange in the beginning. That's because we are not used to specific quantum effects in our everyday lifes (despite the apparently trivial fact that matter is quite stable).

1) I red that the wavefunction of an electron is spread all over the universe, so there's a probability that we can find the electron basically everywhere. How do physicists do the measurement to determine the position of an electron since on the common sense view it seems that they should eliminate every portion of space without the electron to find it? Any normal description?

You must not forget, that quantum theory, as any theory in physics, is just our description of observations. Of course, in nature there are no "wave functions" or even more abstract "Hilbert space vectors", but the wave function is just our description of a particle, and it's meaning is probabilistic, i.e., if the particle is described by the pure quantum state represented by the wave function ##\psi(t,x)##, the probability density (probability per volume) to observe the particle at time ##t## at a position is given by ##P(t,x)=|\psi(t,x)|^2##. You just use any appropriate detector to measure the electron, e.g., some scintillator or pixel detector which tells you that at some time ##t## an electron hit the detector material in a (more or less well defined) region. The position resolution of any detector is always finite, but can be (in principle) as accurate as you like.

2) How do we know that electrons interact with other stuff in everyday life (like air) since they can be anywhere so we can't specify the place of the interaction and by that the interaction itself?

As stressed above, the electron's location is determined by its interaction with the detector.

3) I also red that the interaction between the electron and other things causes the electron to decohere, and serves as a measurement, so how does this 'real life' measurement differ from the measurement in the lab?

In nothing. The measurement devices in the lab are part of nature and thus are not different from observabing something in "real life". It's usually much more accurate than our everyday observations.

4) Does the electron probability cloud reduce when in interaction with other stuff, so for instance can I say that an electron from the atom in my desk can now be in, let's say, South Africa? It seems pretty confusing.

That will tricker a lot of debate on the interpretation of quantum theory. So to a certain extent the answer to this question may differ from one physicist to another. I'm a proponent of the socalled "minimal statistical interpretation", and one has to take into account that the most comprehensive model about particles is relativistic quantum field theory. So an instantaneous collapse as assumed by some other flavors of the Copenhagen interpretation contradicts the basic principles of relativistic QFT, which is based on the assumption of local interactions and microcausality, i.e., all interactions are local and any interaction cannot causally act over space-like distances (i.e., with a faster-than light speed).

5)What about the nucleus? In the modern atomic model it is said that the electron doesn't have a definite position, but the nucleus is regarded as almost fixed in position and momentum. What is the uncertainty in position of the nucleus and how can we know where the atom as a whole is?

Any massive object is subject to the Heisenberg uncertainty relation, ##\Delta x \Delta p \geq \hbar/2##, no matter how big it is. So the nucleus or the entire atom have always at least this minimal uncertainty of position and momentum without any exception. As with any other observable we know where the atom as a whole is from a corresponding measurement or preparation procedure.

Again, I know that this questions may seem weird but I'm a beginner so thanks in advance for the patience, I hope that some explanation will clear up the mess in my head.

The questions are not weird at all, but unfortunately many QT textbooks present QT as something weird (we have currently two threads about this). My tip is: First learn the formalism and adopt the "shutup-and-calculate interpretation" (which in some sense is the minimal statistical interpretation). As soon as you get into discussions about "interpretation" it becomes weird, and I guess, you won't find two physicists following exactly the same "interpretation".

 

[UncertaintyAjay]

What mfb means is that 10^-30mm is absurdly tiny. So it makes no sense to say you can measure the position of a desk to that kind of accuracy. In context of your quote, basically, the author of whatever text this is is saying that for macroscopic objects any uncertainty arising from the uncertainty priniciple is negligible.

 

[durant35]

What mfb means is that 10^-30mm is absurdly tiny. So it makes no sense to say you can measure the position of a desk to that kind of accuracy. In context of your quote, basically, the author of whatever text this is is saying that for macroscopic objects any uncertainty arising from the uncertainty priniciple is negligible.

Ok, thanks to you and mfb, I also thought the same. But how can we know that the electron interacts with anything if we don't know where it is located?

 

[UncertaintyAjay]

The Uncertainty Principle does not say that we do not know where the electron is located. What it says is if we have an electron and we try to measure its position, the more accurately we measure its position, the greater our uncertainty about its momentum and vice-versa.
You can still know if an electron interacts with something. An electron interacting with a photon will give off a flash. It's just that in doing so, your uncertainty of the electron's momentum increases.

 

[durant35]

So let me get this straight and try to get to a normal conclusion, the greater the interaction between the quantum entity and its environment, the lesser possible positions can the entity have? Does the increased momentum result in more certainty in position?

 

[UncertaintyAjay]

No.
The uncertainty principle does not talk about the actual value of position or momentum. It merely says this:
The position and momentum of something cannot be simultaneously measured with arbitrarily high accuracy.
It is a statement about the accuracy with which it is possible to measure momentum and position. If you measure one to a high accuracy, your measurement of the other must necessarily be more inaccurate. The relationship between the error in measurement of momentum ( Δp) and error in position ( Δx) are related to each other by:
ΔpΔx≥h/2π ( equation 1)
where h is Planck's constant ( 6.63 * 10^-34). Let's say your measurement of the electron's position is fairly accurate and the error is tiny. Then the error in position is necessarily larger than h/(2π*Δx). I.e:
Δp≥h/(2π*Δx) ( equation 2).
So if your error in measurement of position is small, you can see from equation 2 that error in measurement of position is large.

So increased momentum does not result in more certainty in position.

It is also important to note that the uncertainty principle does not talk about the accuracy of your apparatus. I could be using the most accurate apparatus ever and this law would still apply. Try as hard as you want. The uncertainty principle is inescapable. The reason for this is because the very act of measurement interferes with the system and changes a state.

For example if you wanted to measure the position of an electron, you could do it by having a light source and observing a flash as the electron goes by. But the interaction between the electron and the photon will result in a change in the momentum of the electron. So the act of measuring changes the system. Only, in macroscopic systems this phenomenon is so small it can easily be neglected.

 

[UncertaintyAjay]

Also, Quantum Mechanics by its very nature is weird ( which makes it so much more interesting). So there is such thing as a normal conclusion, only a logical one.

 

[mfb]

But how can we know that the electron interacts with anything if we don't know where it is located?

If electrons would not interact with anything, we would not have electricity - even worse, we would not have any atoms. So clearly they do interact with other things.
Do you have to know the position of every atom in your hand with sub-nanometer precision to write posts? No.

So let me get this straight and try to get to a normal conclusion, the greater the interaction between the quantum entity and its environment, the lesser possible positions can the entity have?

Not necessarily, it depends on the interaction.

Does the increased momentum result in more certainty in position?

No, at least not in general.

 

[durant35]

No.
The uncertainty principle does not talk about the actual value of position or momentum. It merely says this:
The position and momentum of something cannot be simultaneously measured with arbitrarily high accuracy.
It is a statement about the accuracy with which it is possible to measure momentum and position. If you measure one to a high accuracy, your measurement of the other must necessarily be more inaccurate. The relationship between the error in measurement of momentum ( Δp) and error in position ( Δx) are related to each other by:
ΔpΔx>h/2π ( equation 1)
where h is Planck's constant ( 6.63 * 10^-34). Let's say your measurement of the electron's position is fairly accurate and the error is tiny. Then the error in position is necessarily larger than h/(2π*Δx). I.e:
Δp>h/(2π*Δx) ( equation 2).
So if your error in measurement of position is small, you can see from equation 2 that error in measurement of position is large.

So increased momentum does not result in more certainty in position.

It is also important to note that the uncertainty principle does not talk about the accuracy of your apparatus. I could be using the most accurate apparatus ever and this law would still apply. Try as hard as you want. The uncertainty principle is inescapable. The reason for this is because the very act of measurement interferes with the system and changes a state.

For example if you wanted to measure the position of an electron, you could do it by having a light source and observing a flash as the electron goes by. But the interaction between the electron and the photon will result in a change in the momentum of the electron. So the act of measuring changes the system. Only, in macroscopic systems this phenomenon is so small it can easily be neglected.

Okay, thank you for the example. I just don't understand why is the "greater or equal" sign in the calculation for the uncertainty. Why it isn't just equal so a standard deviation in momentum corresponds to an exact value in the standard deviation of position?

 

[UncertaintyAjay]

Because the actual error would actually depend on your apparatus. If I very measure momentum and then use inaccurate apparatus to measure position, Δx will be greater than h/4π because its very inaccurate apparatus. But no matter how precise my equipment for measuring either of the two quantities, ΔpΔx simply will not be less than h/4π.

Edit: On checking it's actually h/4π not h/2π and I've changed that in my posts. BUT, the points I have made still stand.

 

[entropy1]

If you will grant me an intrusion in this thread with one question, Durant35 ,

Do you have to know the position of every atom in your hand with sub-nanometer precision to write posts?

I was wondering what the interpretation of the word "know" would be in this context, mfb!

 

[mfb]

"Know" as in "can you write the position down (in suitable coordinates)?"

 

[entropy1]

"Know" as in "can you write the position down (in suitable coordinates)?"

I was wondering (for I don't know ) if there has to be a macro-system involved to fix the value of the measurement, and if perhaps there has to be decoherence involved to fix the value?

 

[mfb]

I don't see how this would be related to the point of my statement.

 

[entropy1]

Rephrased: where in the measurement setup becomes the value of the measurement definite?

(Sorry for intruding - last question!)

 

[mfb]

That depends on your favorite interpretation of quantum mechanics.

 

[vanhees71]

Because the actual error would actually depend on your apparatus. If I very measure momentum and then use inaccurate apparatus to measure position, Δx will be greater than h/4π because its very inaccurate apparatus. But no matter how precise my equipment for measuring either of the two quantities, ΔpΔx simply will not be less than h/4π.

Edit: On checking it's actually h/4π not h/2π and I've changed that in my posts. BUT, the points I have made still stand.

As you see, the claim that one cannot measure position and momentum more precisely than given by the uncertainty relation is misleading. Correct is to say that you cannot determine position and momentum of a system in a way that violates the uncertainty relation.

The point is that the quantum mechanical state refers to probability distributions for position, momentum, and any other sensible observable of the system. You can measure either position or momentum at arbitrary precision. To verify the uncertainty relation, you have to measure them more precisely than the standard deviation of these quantities due to the preparation of the system. You can do this by measuring the position of a large ensemble of equally prepared particles and also the momentum of (another) large ensemble of equally prepared particles. That's the meaning of the probabilistic nature of the quantum mechanical state.

 

[durant35]

I have another question. Considering the double slit experiment, during the passing of the electron through both slits, it seems that the electron can be found only in the areas under the obstacle (and not in the obstacle itself). How is that represent by the Heisenberg uncertainty principle, is the obstacle area completely excluded from the probability area so that we can't find the electron in the obsacle and does the electron split the probability areas in two portions of space during the passing through the obstacle?

 

[durant35]

And a sub question, can somebody show me by an example how is the uncertainty principle negligible when considering macroscopic objects? By an example and the equation, for instance if we take a small number for the standard deviation of the momentum

 

[BvU]

And a sub question, can somebody show me by an example how is the uncertainty principle negligible when considering macroscopic objects? By an example and the equation, for instance if we take a small number for the standard deviation of the momentum

Check out the Broglie wavelength for anything on an observable scale. A dust particle or something. And for a baseball...

 

[BvU]

I have another question. Considering the double slit experiment, during the passing of the electron through both slits, it seems that the electron can be found only in the areas under the obstacle (and not in the obstacle itself). How is that represent by the Heisenberg uncertainty principle, is the obstacle area completely excluded from the probability area so that we can't find the electron in the obsacle and does the electron split the probability areas in two portions of space during the passing through the obstacle?

Check out the Feynman lectures on this subject (quite a different link from the one in post #2...)

 

[UncertaintyAjay]

Check out the Feynman lectures on this subject (quite a different link from the one in post #2...

Literally the best source to learn this stuff( the best source that I have come across, which isn't much, but it's Feynman so I have faith that its one of the best ones out there.)

 

[durant35]

@BvU thanks for the article, I will check it out.

Right now, I'm stuck in the corelation between de Broglie wavelength and Uncertainty principle. I've red that the wavelength for big objects is enormously small so their wave properties are negligible. I would like someone to show me an example how are wavelengths connected to the equation of the Heisenberg principle and how does it look for a hypotethical macroscopic object when we consider standard deviations and why is it negligible

 

[BvU]

Are you familiar with the Fourier transform ?

 

[mfb]

I have another question. Considering the double slit experiment, during the passing of the electron through both slits, it seems that the electron can be found only in the areas under the obstacle (and not in the obstacle itself). How is that represent by the Heisenberg uncertainty principle, is the obstacle area completely excluded from the probability area so that we can't find the electron in the obsacle and does the electron split the probability areas in two portions of space during the passing through the obstacle?

Many particles will be detected at the barrier (doesn't have to be a dedicated detector there), those do not participate in the double-slit experiment.

 

[durant35]

Are you familiar with the Fourier transform ?

No mate unfortunately I'm not... How is it related to uncertainty and my question?

 

[BvU]

A wave packet with a sharp distribution in ##\bf k## (##{\bf p} = \hbar {\bf k}##) has a wide distribution in ##x##: they are each other's Fourier transform (http://www.fisica.net/quantica/Griffiths%20-%20Introduction%20to%20quantum%20mechanics.pdf page 107).

 

[durant35]

When physicists put an electron in a lab or a box, does that mean that there is a 100 percent change that the electron is in the lab or are electrons capable of being even outside of the lab when we put them inside (in a range of 10 m size let's say)? This is connected with my question about the double slit, if we have a small wall and two slits do electrons treat the wall like a barrier or do they go through the wall and also through the slits?

 

[BvU]

You are referring to idealized boundary conditions that feature in exercises for beginning students. Experimental reality is a bit tougher, but still the results from these simple exercises often hold to within all practical limits.

The exercises are really useful.

And there is no difference between 100 percent and 100 - 10^-659 percent.PS did you read up on the Broglie wavelength ?