Understanding Orbital Shapes: The Probability of Electron Location

[Meson080]

Simple pictures showing orbital shapes are intended to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found. The diagrams cannot, however, show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space.

Is the statement by wikipedia correct?

Since, there is a probability of finding electron at any distance from the nucleus, when the electron comes far from the nucleus, I will block it, so that it won't return to its parent atom. Am I not stealing the electron? I can steal even the electron of your body being in India, be careful!

That's what we layman think from those statements. What's the actual meaning of the wikipedia statement?

 

[Drakkith]

It means that if we perform a measurement to find out where the electron is, its location could technically be almost anywhere, including in India. But the probability of finding the electron further than about a nano-meter from the nucleus is so low that you could perform this measurement every second for a billion years and not find it there.

 

[Meson080]

It means that if we perform a measurement to find out where the electron is, its location could technically be almost anywhere, including in India.

Then I do have the chance of stealing your body's electron. Is that what you mean?

But the probability of finding the electron further than about a nano-meter from the nucleus is so low that you could perform this measurement every second for a billion years and not find it there.

Can I have the source for this?

 

[Drakkith]

Have you checked the list of references at the bottom of the wikipedia article you quoted from?

 

[Meson080]

Have you checked the list of references at the bottom of the wikipedia article you quoted from?

It will be helpful, if you can point the source among that bunch of reference links.

 

[Meson080]

The same question is also posted in Physics Stack Exchange. Interested folks can read this page: Can I steal your electron? The page might help to have better discussion.

 

[Drakkith]

I don't have a specific source, it's just general knowledge how atomic orbitals work. My response wasn't meant to be taken literally, as I haven't done the math. I just know that the probability of an electron being found a few thousand miles away from its atom is exceedingly low. So low that we never worry about objects falling apart because they lose their electrons in this manner.

 

[Dale]

It will be helpful, if you can point the source among that bunch of reference links.

See
http://en.m.wikipedia.org/wiki/Hydrogen_atom#Wavefunction

For the ground state electron this simplifies to a probability density of:
$$|\Psi(r)|^2 = \frac{1}{a_0^3 \pi} e^{-2r/a_0}$$

Since $a_0=5.29 \; 10^{-11} \; m$ if you want to steal an electron in a 1 m cubic box located even just 10 m away, the probability is so small that it cannot be distinguished from 0 with even a million digits of precision, and the probability of finding it anywhere in the universe further than 1 m distance away is less than 1.6E-16419451091

 

[Khashishi]

Meson, you have a very, very, very small chance. This chance is really too small to worry about in any context.

 

[Meson080]

I don't have a specific source, it's just general knowledge how atomic orbitals work. My response wasn't meant to be taken literally, as I haven't done the math. I just know that the probability of an electron being found a few thousand miles away from its atom is exceedingly low. So low that we never worry about objects falling apart because they lose their electrons in this manner.

So, once electron comes far from its parent atom, it won't return to it? Did you mean this?

 

[Drakkith]

I believe that's how it works.

 

[Meson080]

So, once electron comes far from its parent atom, it won't return to it? Did you mean this?

I believe that's how it works.

The question is, what provides energy to the electron to go any far distance from the nucleus?

As there is the force which is holding the electron, it should not have any "probability" of going far from the nucleus, isn't it? How does QM tackle this discrepancy?

 

[Nugatory]

The question is, what provides energy to the electron to go any far distance from the nucleus?

To measure the position of the electron you have to interact with it, and that interaction supplies any necessary energy. The total energy of the system (nucleus, electron, and measuring device) is conserved.

Understand also that the electron isn't anywhere until you interact with it. The function that DaleSpam posted does not give you the probability that the electron is at a given location, it gives you the probability that the electron will be found at that location if you make a measurement. Thus, there's no question about how the electron moved far away from the nucleus before you looked and found it out there - until you measured its position it didn't have a position, it wasn't far away from the nucleus, or near it, or anywhere else.

That's how quantum mechanics works. If you don't like it, you're in good company - but like it or not, them's the rules.

 

[Drakkith]

I believe energy conservation in QM is a bit more complicated than it is in classical physics, but you'd need to ask in the QM forum if you want to know about that.

 

[Meson080]

Understand also that the electron isn't anywhere until you interact with it. The function that DaleSpam posted does not give you the probability that the electron is at a given location, it gives you the probability that the electron will be found at that location if you make a measurement. Thus, there's no question about how the electron moved far away from the nucleus before you looked and found it out there - until you measured its position it didn't have a position, it wasn't far away from the nucleus, or near it, or anywhere else.

I felt this as the misconception of Heisenberg's Uncertainity principle. Isn't this?

That's how quantum mechanics works. If you don't like it, you're in good company - but like it or not, them's the rules.

If it works that way, I need to learn more to unlearn as Feynman always says.

 

[phinds]

I felt this as the misconception of Heisenberg's Uncertainity principle. Isn't this?

No, they have nothing do do with each other. That article is about the false presentation of the HUP as a measurement problem. It is not and never has been a measurement problem, it is a fundamental fact of nature. That presentation was a dumbed-down common language presentation that does not represent the math.

If it works that way, I need to learn more to unlearn as Feynman always says.

As did we all when we first got into this stuff.


EDIT: just to be sure I'm clear, when I say "they have nothing to do with each other", I'm saying that the fact that an electron has a probability distribution that gives a non-zero (but incredibly tiny) result for positions far away from its atom has nothing to do with the HUP.

 

[Dale]

The question is, what provides energy to the electron to go any far distance from the nucleus?

The formula I provided is for the ground state, meaning that it has the minimal amount of energy and is not excited. No energy is required for the electron to be measured in different locations in the ground state. Energy is only required to raise it to a different state.

 

[Nugatory]

The formula I provided is for the ground state, meaning that it has the minimal amount of energy and is not excited. No energy is required for the electron to be measured in different locations in the ground state. Energy is only required to raise it to a different state.

OP is capturing the electron ("stealing" in the thread title). The state in which the electron has been localized at some distance from the nucleus isn't an energy eigenstate.

 

[Meson080]

Understand also that the electron isn't anywhere until you interact with it. The function that DaleSpam posted does not give you the probability that the electron is at a given location, it gives you the probability that the electron will be found at that location if you make a measurement.

Can I have any reliable source which supports this idea?

 

[Nugatory]

Can I have any reliable source which supports this idea?

Any decent QM textbook will work. To see it in modern terms you'll want one that stresses the statistical interpretation, but the idea that it makes no sense to talk about the value of quantities that haven't been measured goes all the way back to Bohr.

 

[pbuk]

You can steal a few million of my electrons by brushing your hand across my sweater, it's no big deal. I'll get them back later.

 

[Dale]

OP is capturing the electron ("stealing" in the thread title). The state in which the electron has been localized at some distance from the nucleus isn't an energy eigenstate.

Correct, after the measurement of its position it is a position eigenstate, by definition.

 

[Meson080]

I believe energy conservation in QM is a bit more complicated than it is in classical physics, but you'd need to ask in the QM forum if you want to know about that.

I have exam tomorrow, I will try to post a new question in QM forum within two days. But, let the discussion go on, so that we can have a well built query in QM forum.

 

[Meson080]

You can steal a few million of my electrons by brushing your hand across my sweater, it's no big deal. I'll get them back later.

We are planning to steal electrons sitting far from you. We have missile ideas and not gun ideas like your's!

 

[Meson080]

Understand also that the electron isn't anywhere until you interact with it. The function that DaleSpam posted does not give you the probability that the electron is at a given location, it gives you the probability that the electron will be found at that location if you make a measurement. Thus, there's no question about how the electron moved far away from the nucleus before you looked and found it out there - until you measured its position it didn't have a position, it wasn't far away from the nucleus, or near it, or anywhere else.

Just to make sure I understand you better, do you agree this statement: There is a non-zero probability of finding an electron at any distance from the nucleus, even if we don't make a measurement. By "measurement" I mean, using photons to know the address of electron.

 

[phinds]

I don't think it could be outside the light cone of the electron, so "any distance" is too much even if the rest is correct (and I'm not saying it is or isn't, although I THINK that it is).

 

[Dale]

There is a non-zero probability of finding an electron at any distance from the nucleus, even if we don't make a measurement.

No. If you don't make a measurement then you have exactly 0 probability of finding it anywhere. You cannot find something if you don't look.

 

[Nugatory]

Just to make sure I understand you better, do you agree this statement: There is a non-zero probability of finding an electron at any distance from the nucleus, even if we don't make a measurement. By "measurement" I mean, using photons to know the address of electron.

There might be other ways of locating the position of the electron... But if you don't interact with it in some way you have no way of finding it anywhere.

 

[Nugatory]

I don't think it could be outside the light cone of the electron, so "any distance" is too much even if the rest is correct (and I'm not saying it is or isn't, although I THINK that it is).

Where are you putting the apex of that light cone? Basic first-quantization QM is neither relativistic nor local and you can plug arbitrarily large values of r into the spherical harmonics.

QED is relativistic, but it doesn't let you talk about "the" electron at all.

 

[phinds]

Where are you putting the apex of that light cone? Basic first-quantization QM is neither relativistic nor local and you can plug arbitrarily large values of r into the spherical harmonics.

QED is relativistic, but it doesn't let you talk about "the" electron at all.

OK, I'll take your word for it. Thanks.

 

[Meson080]

No. If you don't make a measurement then you have exactly 0 probability of finding it anywhere. You cannot find something if you don't look.

So, atom has definite boundary if we don't try to make measurement i.e if we don't try to see the atom. Is this what you mean?

 

[Dale]

No, that is not at all what I said. If you don't make a measurement then you have a 0 probability of finding the electron anywhere, including very close to the nucleus. There is no definite boundary whatsoever.

If you want to find the electron anywhere you must measure its location. In a fundamental sense quantum objects do not have any definite value of a property until that property is measured.

 

[Meson080]

No, that is not at all what I said. If you don't make a measurement then you have a 0 probability of finding the electron anywhere, including very close to the nucleus. There is no definite boundary whatsoever.

Even if we don't make measurement, there will be electron in the atom, I hope this is obvious. This means, there is a probability of finding electron anywhere, at least very close to the nucleus, even if don't make measurement.

 

[Nugatory]

Even if we don't make measurement, there will be electron in the atom, I hope this is obvious. This means, there is a probability of finding electron anywhere, at least very close to the nucleus, even if don't make measurement.

You are still trying to hold on to the idea that the electron has a position before it is measured. It doesn't.

The quantum mechanical wave function gives the probability that a measurement will give a particular result, but says nothing about what's going on if no measurement is made. It seems very natural to assume that if a measurement of the electron position would give us a particular position, then the electron must really be in that position whether we look or not... But that's an additional assumption, and one that turns out not to be valid.

 

[voko]

"finding" is "making a measurement". Grasp this, or you will never understand the rest.

 

[Dale]

Even if we don't make measurement, there will be electron in the atom, I hope this is obvious. This means, there is a probability of finding electron anywhere, at least very close to the nucleus, even if don't make measurement.

Finding electron = measurement. If you don't measure then P(measurement)=0 and therefore P(finding electron)=0. This is true even classically.

QM goes beyond that. In QM the electron does not even have a definite position until its position is measured. The only time that you can assert that a QM mechanical system has a definite property without measuring it is if the system is in an eigenstate of the corresponding operator. The ground state is not an eigenstate of position.

 

[Meson080]

Ok, let's assume that we have a girl moving randomly over a certain area, with a elastic rope tied to her hand, assume that the other end of the rope is tied to the nail. The nail is fixed at the center of the area over which she moves randomly. Compare this situation with our electron, where electron is the girl, and the flexible rope refers to the force which holds electrons in the atom.

Now, assume that we are all standing in our countries closing our eyes, trying to catch her and give her freedom, everytime waving our hands, would the girl reach us? If she reaches, will the rope gets cutted off?

For catching the electron, let's say we have the field trapper, which exerts electric field on the electron to steal it.

Sorry, if I am asking the same question again and again, I think this will allow me to understand better. What would you say, do we get her or not?

 

[Nugatory]

Compare this situation with our electron, where electron is the girl, and the flexible rope refers to the force which holds electrons in the atom.

The two situations are not comparable. The girl and the flexible rope consist of something on the order of 10²⁵ particles, all of which are constantly interacting with one another. Because they are always interacting you will never find any significant number of them, let alone all them, in an energy eigenstate where energy is certain but position is not. That's exactly opposite from how it is for a few electrons around an atomic nucleus.

You could try googling for "quantum decoherence"; this is the process by which systems composed of many particles collectively behave according to our classical intuition even though each one of the individual particles is behaving according to the Rules of Quantum Mechanics. Or, if you find the math a bit heavy going, you could try the pretty decent non-technical explanation in a book called "Where does the weirdness go?". It's available through Amazon: https://www.amazon.com/dp/0465067867/?tag=pfamazon01-20

 

[Dale]

I agree. The girl scenario is completely irrelevant to the electron scenario. Electrons don't behave like girls and the electric field doesn't behave like an elastic rope. You cannot learn anything about one by considering the behavior of the other.

 

[Meson080]

I agree. The girl scenario is completely irrelevant to the electron scenario. Electrons don't behave like girls and the electric field doesn't behave like an elastic rope. You cannot learn anything about one by considering the behavior of the other.

In layman terms, can you explain the difference between the two scenarios?

 

[Nugatory]

In layman terms, can you explain the difference between the two scenarios?

In layman's terms, the girl consists of about 10²⁵ interacting particles whereas the electron in the potential of a hydrogen nucleus is a single particle.

 

[voko]

I would put it differently. An electron is quantum mechanics, a girl is classical mechanics. It is not possible to interpret quantum mechanics via classical mechanics, that simply does not work.

 

[Meson080]

Ok, let's leave the girl . We don't want her, atleast for you all.

Consider the same situation, where we all are standing in our own countries, closing our eyes, with the "electron trappers" in our hands, to to trap the electron (assume that they use electric field to trap electron as said before). Will the electron of the atom from an alien world's alien, ever come and gets trapped into the trapper? If it reaches the trapper, does the electron losses the influence of the atom? If we don't trap it, will it return to its parent?

 

[voko]

If an electron gets trapped, it further evolution depends primarily on the "trapper". If no electron is trapped, then we have no information at all (unless our experiment is more complex and we have other detection devices).

 

[Dale]

In layman terms, can you explain the difference between the two scenarios?

I think that Nugatory's answers did that pretty well. The only other thing I would mention is that in classical mechanics the state space is not a vector space while in quantum mechanics it is. This is the root of most of the quantum weirdness.

 

[Dale]

Will the electron of the atom from an alien world's alien, ever come and gets trapped into the trapper?

Probably not in our lifetime.

If it reaches the trapper, does the electron losses the influence of the atom? If we don't trap it, will it return to its parent?

Once you have measured it to be in any position the wavefunction will collapse to the corresponding eigenstate of the position. This was mentioned by Nugatory in post 18. Such an eigenstate is no longer the ground state, so the electron is no longer bound to the nucleus, however, such a state is also not a stationary state, and the wavefunction will evolve over time. How it evolves probably depends more on the design of the box than on the parent nucleus.

 

[Meson080]

Probably not in our lifetime.

Once you have measured it to be in any position the wavefunction will collapse to the corresponding eigenstate of the position. This was mentioned by Nugatory in post 18. Such an eigenstate is no longer the ground state, so the electron is no longer bound to the nucleus, however, such a state is also not a stationary state, and the wavefunction will evolve over time. How it evolves probably depends more on the design of the box than on the parent nucleus.

See (you are also expected to be blind in this experiment ) I don't know whether the electron comes near me or not, I am closing eyes (I am blind!) , with trapper in my hand, waving it every time. Suppose it reaches and I will not catch it, assume that I will not be knowing whether it came near me or not, will it return to its parent?

I asked this question in my last post itself, I felt this question was not answered.

 

[phinds]

See (you are also expected to be blind in this experiment ) I don't know whether the electron comes near me or not, I am closing eyes (I am blind!) , with trapper in my hand, waving it every time. Suppose it reaches and I will not catch it, assume that I will not be knowing whether it came near me or not, will it return to its parent?

I asked this question in my last post itself, I felt this question was not answered.

Yes, it HAS been answered. If you don't catch it you have NO information and cannot "assume" that it was anywhere near you.

 

[Nugatory]

See (you are also expected to be blind in this experiment ) I don't know whether the electron comes near me or not, I am closing eyes (I am blind!) , with trapper in my hand, waving it every time. Suppose it reaches and I will not catch it, assume that I will not be knowing whether it came near me or not, will it return to its parent?

In quantum mechanics, as "observation" is any irreversible interaction with the environment. It is irrelevant whether the environment includes a conscious observer looking to see whether there was an interaction.

Does the electron interact with the trapper or does it not? If it does, the electron has been detected in the location defined by the trapper, whether someone looks or not. If it does not, then there is no detection and the electron still has no position.

 

[Dale]

Suppose it reaches and I will not catch it, assume that I will not be knowing whether it came near me or not, will it return to its parent?

I asked this question in my last post itself, I felt this question was not answered.

What exactly do you feel was unanswered? I specifically stated that it is not bound to the nucleus.

Whether or not you are blind or choose not to look at the answer is not relevant, as others have mentioned. What matters is whether or not the experimental apparatus interacts with the electron in such a way as to measure the position.