# Where is the electron?

To make myself clearer, lets imagine the same apparatus used by the Schrodinger Cat Paradox(without the cat and the poison). An electron is shot from position A. It is moving to the other side of the box.
Now, the quesion is, when we open the box and observe the electron after a long period, where will the electron be?
Is it:
At the other side of the box?
Position A?
At the double slit?

chroot
Staff Emeritus
Gold Member
There isn't enough information in the question to actually formulate an answer, but I assume you're asking about how a wavefunction evolves over time.

The answer is that a wavefunction spreads out over time. After a sufficiently long period of time, the electron has an equal chance of being anywhere in the box.

- Warren

SpaceTiger
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Gold Member
chroot said:
The answer is that a wavefunction spreads out over time. After a sufficiently long period of time, the electron has an equal chance of being anywhere in the box.
Are you sure? If the box is being treated as an infinite potential well, then there are steady-state solutions to the SE, so its wave function depend on the energy of the state and the size of the box. If it's not treated as infinite, then given enough time, it seems like it would tunnel out of the box.

ZapperZ
Staff Emeritus
SpaceTiger said:
Are you sure? If the box is being treated as an infinite potential well, then there are steady-state solutions to the SE, so its wave function depend on the energy of the state and the size of the box. If it's not treated as infinite, then given enough time, it seems like it would tunnel out of the box.
Oy...

The "box" is HUGE when compare to the size of the electron. When you get to that scale, it is quite unproductive to treat this as a QM problem. Particle accelerators don't. Besides, why would this matter? The "box" is simply there as an opaque barrier to prevent the cat being observed (and presumably, the poison from escaping and killing the potential observer also). It is not meant to be a potential barrier.

Zz.

Sterj
I have a question on you physiscists:

We know that an electron is in the middle of the box (by measuring). After a time lets take 10 seconds this electron can be everywhere.

Is that sadisfying you? I mean it's absurd that we can't really calculate where this thing is. Perhaps there are to many other objects that have influence on the electrons path.

ZapperZ
Staff Emeritus
Sterj said:
I have a question on you physiscists:

We know that an electron is in the middle of the box (by measuring). After a time lets take 10 seconds this electron can be everywhere.

Is that sadisfying you? I mean it's absurd that we can't really calculate where this thing is. Perhaps there are to many other objects that have influence on the electrons path.
Say what?

What box are we talking about here? How did you "measure" it's position?

I have an electron linear accelerator. I can tell when a bunch of electrons is passing through a particular location (via an Integrated Charge Transformer device). These electrons are NEVER "everywhere" after that. I can detect them further down the beamline later on. They behave just like classical particles.

Or are you misapplying QM, and in particular, the HUP, here?

Zz.

jtbell
Mentor
ZapperZ said:
The "box" is HUGE when compare to the size of the electron.
Depends on the size of the box, of course. A crucial detail which was not explicitly stated in the original question. Of course, a box which is big enough to contain Schrödinger's Cat surely meets your description, unless Schrödinger had a very tiny cat. :rofl:

ZapperZ
Staff Emeritus
jtbell said:
Depends on the size of the box, of course. A crucial detail which was not explicitly stated in the original question. Of course, a box which is big enough to contain Schrödinger's Cat surely meets your description, unless Schrödinger had a very tiny cat. :rofl:
A nano-cat..... hum.... I see research funding opportunities here! :)

Zz.

chroot
Staff Emeritus
Gold Member
Yes, I made some tacit assumptions in my previous answer, which I perhaps should have expressed clearly:

1) I assume that the box is extremely large, thus having an essentially infinite number of energy states for the electron.

2) I assume the box is full of air, thus providing decoherence in the remarkable possibility that the electron was originally in a stationary state.

- Warren

chroot said:
...I assume that the box is extremely large, thus having an essentially infinite number of energy states for the electron. - Warren
Size of box determines the spacing between the possible energy levels, not the number. Smaller box, wider gap between level n and n+1.

chroot
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Billy T said:
Size of box determines the spacing between the possible energy levels, not the number. Smaller box, wider gap between level n and n+1.
True. I spoke sloppily. Any box has an infinite number of energy levels; I meant that a large box has an infinite number of essentially macroscopically indistinguishable energy levels, a continuum.

- Warren

SpaceTiger
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Gold Member
ZapperZ said:
The "box" is HUGE when compare to the size of the electron. When you get to that scale, it is quite unproductive to treat this as a QM problem.
Well, he didn't specify the box as being huge, but it shouldn't matter, as the SE would still be an appropriate physical description. chroot is right, however, in asserting that the presence of air would change the situation.

Particle accelerators don't. Besides, why would this matter?
Particle accelerators certainly do consider quantum mechanics. I'm not sure what you mean by this.

If you're saying that they don't use a steady-state solution to the SE for the interior of the accelerator, that's simply because the electron's wave function doesn't have nearly enough time to reach that steady state. In the problem that was stated, I assumed that we were giving the particle's wave function an indefinite amount of time to evolve, in which case it would simply be the solution to a 3-D infinite potential well. If they don't give the particle much time to evolve to that state, then an approximate solution can be obtained classically (that is, you just follow the particle's sequence of collisions with the box walls and figure out where it is after some time) and the issue of probability can be ignored.

The "box" is simply there as an opaque barrier to prevent the cat being observed (and presumably, the poison from escaping and killing the potential observer also). It is not meant to be a potential barrier.
I was addressing his question about the electron's wave function. It had nothing to do with the cat and the poison.

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ZapperZ
Staff Emeritus
SpaceTiger said:
Well, he didn't specify the box as being huge, but it shouldn't matter, as the SE would still be an appropriate physical description. chroot is right, however, in asserting that the presence of air would change the situation.
Why wouldn't the size here matters? At some point, the size would be enough to consider the electrons being free, classical particles. See below.

Particle accelerators certainly do consider quantum mechanics. I'm not sure what you mean by this.

If you're saying that they don't use a steady-state solution to the SE for the interior of the accelerator, that's simply because the electron's wave function doesn't have nearly enough time to reach that steady state. In the problem that was stated, I assumed that we were giving the particle's wave function an indefinite amount of time to evolve, in which case it would simply be the solution to a 3-D infinite potential well. If they don't give the particle much time to evolve to that state, then an approximate solution can be obtained classically (that is, you just follow the particle's sequence of collisions with the box walls and figure out where it is after some time) and the issue of probability can be ignored.
If you look at the subject of Beam Physics, you will see that in practically ALL cases, the dynamics of an electron beam is described by CLASSICAL mechanics and E&M, not QM description. I'm not going to argue about this because you can double-check this yourself to verify what I have said. Simulation softwares that people in this field use to track particles in the beam pipe such as PARMELA and PIC codes ALL use classical physics, never QM. These are classical particles, no longer QM particles.

Zz.

SpaceTiger
Staff Emeritus
Gold Member
ZapperZ said:
Why wouldn't the size here matters? At some point, the size would be enough to consider the electrons being free, classical particles. See below.
The size matters for timescales. The initial conditions for the electron are based on our measurements its origins. Relative to the size of a human-scale box, this is practically a delta function. However, given enough time, the particle will approach a steady-state solution to the SE. The larger the box (or accelerator), the longer this time is.

When the particle is shot into the box, it will be basically a delta function and can be described classically, but over very long periods of time, it disperses and approaches the 3-D infinite square well solution. As the box becomes larger relative to the de Broglie wavelength, then this approaches a continuum, as chroot pointed out.

If you look at the subject of Beam Physics, you will see that in practically ALL cases, the dynamics of an electron beam is described by CLASSICAL mechanics and E&M, not QM description. I'm not going to argue about this because you can double-check this yourself to verify what I have said.
Heh, if you're selective about which aspects of the accelerator you're talking about, sure. The collision rates can be described classically, but the collision itself is fundamentally quantum mechanical and you can't describe its output particles without QM.

ZapperZ
Staff Emeritus
SpaceTiger said:
The size matters for timescales. The initial conditions for the electron are based on our measurements its origins. Relative to the size of a human-scale box, this is practically a delta function. However, given enough time, the particle will approach a steady-state solution to the SE. The larger the box (or accelerator), the longer this time is.

When the particle is shot into the box, it will be basically a delta function and can be described classically, but over very long periods of time, it disperses and approaches the 3-D infinite square well solution. As the box becomes larger relative to the de Broglie wavelength, then this approaches a continuum, as chroot pointed out.

Heh, if you're selective about which aspects of the accelerator you're talking about, sure. The collision rates can be described classically, but the collision itself is fundamentally quantum mechanical and you can't describe its output particles without QM.
Collision? COLLISION?

I could have sworn that I am talking about particle ACCELERATOR, and not particle COLLIDER! Where did colliding particle came into this? You do know that those two are not one of the same. I can have a particle accelerator in an electron storage ring that has NOTHING to do with high energy physics of particle collider.

And as far as "time" goes, I would suggest you look at how long an electron bunch is kept "alive" in a synchrotron (try 12 hours at a time or even more!). I can tell you from personal experience and knowledge that none of the beam physics issues associated with a synchrotron beam have any QM in it.

Zz.

SpaceTiger
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Gold Member
ZapperZ said:
I could have sworn that I am talking about particle ACCELERATOR, and not particle COLLIDER! Where did colliding particle came into this? You do know that those two are not one of the same. I can have a particle accelerator in an electron storage ring that has NOTHING to do with high energy physics of particle collider.
Fine, you're right, in an accelerator alone, they don't need to consider QM. In my experience, most accelerators are just parts of colliders, so pardon me for using incorrect terminology.

And as far as "time" goes, I would suggest you look at how long an electron bunch is kept "alive" in a synchrotron (try 12 hours at a time or even more!). I can tell you from personal experience and knowledge that none of the beam physics issues associated with a synchrotron beam have any QM in it.
This is exactly what I said in my first response to you.

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ZapperZ
Staff Emeritus
SpaceTiger said:
Fine, you're right, in an accelerator alone, they don't need to consider QM. In my experience, most accelerators are just parts of colliders, so pardon me for using incorrect terminology.
Well, no, I'm not trying to be difficult here, but MOST accelerators in the world are NOT part of particle colliders. They are more part of synchrotron centers - after all, there are more synchrotrons then there are particle colliders. And there are particle accelerators in medical physics to generate higher intensity and better X-rays for treatment and diagnostic purposes. People who work in particle accelerators often do NOT work in particle physics/colliders. Particle accelerator conferences, such as the upcomming PAC05, is different than the high energy physics conference such as the APS April Meetings.

This is exactly what I said in my first response to you.
Where exactly? Your first response to me was:

"Well, he didn't specify the box as being huge, but it shouldn't matter, as the SE would still be an appropriate physical description."

and

"Particle accelerators certainly do consider quantum mechanics. I'm not sure what you mean by this.

If you're saying that they don't use a steady-state solution to the SE for the interior of the accelerator, that's simply because the electron's wave function doesn't have nearly enough time to reach that steady state. In the problem that was stated, I assumed that we were giving the particle's wave function an indefinite amount of time to evolve, in which case it would simply be the solution to a 3-D infinite potential well. If they don't give the particle much time to evolve to that state, then an approximate solution can be obtained classically (that is, you just follow the particle's sequence of collisions with the box walls and figure out where it is after some time) and the issue of probability can be ignored."

As far as I can tell, you are saying that the LONGER the time period for the electrons are around in the accelerator, the more definite and appropriate and relevant the QM wavefunction is. This NOT what I have said. I have pointed out that in the synchrotron storage rings that can keep electrons for sometime MORE than 12 hours, they do not use QM descriptions. The dynamics of the beam physics are purely classical - the electrons are clearly classical particles being treated via classical E&M.

Again, I would recommend you to look at, for example, Phys. Rev. Special Topic - Accelerators and Beam journal. You will be hard pressed to find any QM description being used in the beam physics applications.

Zz.

SpaceTiger
Staff Emeritus
Gold Member
As far as I can tell, you are saying that the LONGER the time period for the electrons are around in the accelerator, the more definite and appropriate and relevant the QM wavefunction is. This NOT what I have said. I have pointed out that in the synchrotron storage rings that can keep electrons for sometime MORE than 12 hours, they do not use QM descriptions. The dynamics of the beam physics are purely classical - the electrons are clearly classical particles being treated via classical E&M.
That's exactly what I'm saying, but the timescales I'm talking about are much longer than 12 hours. I figured you were telling me that this was too short for the wave function to spread out, which is consistent with what I said in that paragraph.

This is all rather beside the point, however, as the interior of an accelerator isn't even close to being approximated by an infinite square well, like the case we were originally discussing.

ZapperZ
Staff Emeritus
SpaceTiger said:
That's exactly what I'm saying, but the timescales I'm talking about are much longer than 12 hours. I figured you were telling me that this was too short for the wave function to spread out, which is consistent with what I said in that paragraph.
How long of a time scale? The age of the universe?

This is all rather beside the point, however, as the interior of an accelerator isn't even close to being approximated by an infinite square well, like the case we were originally discussing.
We were "arguing" your claim that accelerator physics treatment of the beam dynamics make use of QM, and I dispute that. If you didn't think the interior of an accelerator isn't "even close" to being an infinite square well, then you shouldn't have made that claim in the first place.

I also don't get this "infinite square well" thing. Are you saying that quantum effects can only be clearly manifested when a particle is in an infinite square well? So an accelerator isn't actually LARGE enough to show QM effects? This would be rather puzzling since clear quantum effects would be more clearly manifested when an electron is confined to smaller spatial dimensions.

Zz.

For fun and in order to give some figures rather than long sentences, let’s consider a numerical example (I am not an expert in accelerators, so I will just give some rough numbers):
If we consider a electron in an accelerator between two interactions (e.g. the quadrupole magnets); we may approximate the electron by a free wave packet with a given delta_q,delta_p.

From, e.g., Messiah, quantum mechanics, we have the time evolution of the delta_q of the free wave packet to the first order given by:

Delta_q(t)= delta_q(o) . [ 1+1/4 . (D.Lb/(delta_q(o)) 2)2] ½

Where D is the distance covered by the free electron between two instants (D~v.t) and Lb is the mean wavelength of the electron wave packet ~ hbar/mv. (we neglect the relativistic effects).

Thus if we assume v ~ c; we have Lb ~ 3,86.10^-13m (high energy accelerators).

Therefore if we want a Delta_q(t) somewhat different from delta_q(o), we need:
(D.Lb)½ ~ delta_q(o) (or greater than)
<=> D ~ delta_q(0)2/Lb
Therefore if the quadrupole magnet in the accelerator are able to focus the electron beam into a 1mm diameter tube, we have delta_q(0) ~1mm (at the output of a magnet) and therefore we have D ~ 2600Km (while the size of accelerators ~ km/10km) in order to see some quantum effects on the beam.
That means between two interactions in the accelerator (e.g the quadrupole magnets or other apparatuses), we may neglect the wave packet widening, i.e. we may consider the electron as a classical particle.
However, we need to get some interactions to focus the beam at regular intervals (as well as to control the direction of the beam), so that the electron path between 2 interactions in the accelerator is classical (distance between the interactions << 2600Km in actual accelerators).
(because after 12000Km without interaction we just have delta_q(t) ~ 2 delta_q(0) ~2 mm!).

Seratend.

P.S. There may be some “minor” errors in the above results ; )
P.P.S. Therefore, it is not the age of the universe, but rather the size of the earth that matters ; ).

SpaceTiger
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Gold Member
We were "arguing" your claim that accelerator physics treatment of the beam dynamics make use of QM, and I dispute that. If you didn't think the interior of an accelerator isn't "even close" to being an infinite square well, then you shouldn't have made that claim in the first place.
I conceded that point because I was using the wrong terminology for an accelerator. It was your initially inappropriate analogy that spurned that "argument" in the first place.

I also don't get this "infinite square well" thing. Are you saying that quantum effects can only be clearly manifested when a particle is in an infinite square well?
No, I'm saying that the infinite square well is the case we were originally discussing (for which the solutions are simple). You initially implied that the fact that they don't consider QM in accelerators implies that we should never have to consider it in a large box.

So an accelerator isn't actually LARGE enough to show QM effects? This would be rather puzzling since clear quantum effects would be more clearly manifested when an electron is confined to smaller spatial dimensions.
An accelerator is harder to think about quantum mechanically because it not only has complex potential geometries, but also sometimes time-variable potentials. I couldn't compute the timescales for an electron's wave function dispersal off the top of my head, while the solutions for the infinite square well are well known.

ZapperZ
Staff Emeritus
SpaceTiger said:
No, I'm saying that the infinite square well is the case we were originally discussing (for which the solutions are simple). You initially implied that the fact that they don't consider QM in accelerators implies that we should never have to consider it in a large box.
How does the box corresponds to an "infinite square potential"? Do we know the boundary conditions at the box? A cardboard box? How did we know so much to be able to make that connection to an infinite square potential? How does the size of the box that can contain a typical cat compare to the de Broglie wavelength of an electron typically ejected from a beta decay? Forget about an accelerator, but is the box bigger than, let's say, a typical photoemission analyzer? This is because the photoelectrons that have been emitted from a material is TREATED as classical FREE particles when they go through a typical electron analyzer such as Scienta SES200. I've worked with one, and it is no bigger than a typical "box". Yet, these electrons were NEVER treated as if they're in an "infinite square potential". In fact, if they are, all our photoemission experiements would yield the wrong results!

Zz.

SpaceTiger
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Gold Member
ZapperZ said:
How does the box corresponds to an "infinite square potential"?
We don't. If you'll chill and read my original post, I described the idealized cases I was considering. In the absense of air, it should be a very close approximation to an infinite potential because, as far as an electron is concerned, those box walls are impenetrable. It's not really infinite, however, so eventually it will tunnel out, but not for a very long time (so don't give me any more ridiculous experimental examples).

Do we know the boundary conditions at the box? A cardboard box? How did we know so much to be able to make that connection to an infinite square potential?
Well, it's square and it's a very large potential relative to the electron's energy, so I don't really see why you think it would be a bad approximation.

How does the size of the box that can contain a typical cat compare to the de Broglie wavelength of an electron typically ejected from a beta decay? Forget about an accelerator, but is the box bigger than, let's say, a typical photoemission analyzer? This is because the photoelectrons that have been emitted from a material is TREATED as classical FREE particles when they go through a typical electron analyzer such as Scienta SES200. I've worked with one, and it is no bigger than a typical "box". Yet, these electrons were NEVER treated as if they're in an "infinite square potential". In fact, if they are, all our photoemission experiements would yield the wrong results!
Again, we're talking about much longer timescales than you would see in your experiments. See the discussions of "revival time" http://webphysics.davidson.edu/mjb/wigner/extend_wigner.pdf [Broken]. For wave packets much smaller than the size of the box ($$n_0>>1$$), they will take an extremely long time to exhibit noticable quantum behavior.

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ZapperZ
Staff Emeritus
SpaceTiger said:
We don't. If you'll chill and read my original post, I described the idealized cases I was considering. In the absense of air, it should be a very close approximation to an infinite potential because, as far as an electron is concerned, those box walls are impenetrable. It's not really infinite, however, so eventually it will tunnel out, but not for a very long time (so don't give me any more ridiculous experimental examples).
Well, I'm sorry that I had to support my answer with ACTUAL experimental situations rather than a made-up one. I won't do that again.

Secondly, when an electron is emitted off a material, as in a photoemission process or a beta decay, can you show me how such electrons are treated as far as its description goes? It appears that you are insisting that such an electron can be treated as a QM particle since it can be confined for a "very long time" inside such a device this size. Can you show me one specific example where such a treatment has been done and where this idea is applicable?

Zz.

SpaceTiger
Staff Emeritus
Gold Member
ZapperZ said:
Well, I'm sorry that I had to support my answer with ACTUAL experimental situations rather than a made-up one. I won't do that again.
They weren't ridiculous because they were real, they were ridiculous because they were inappropriate to the scales we were discussing.

Secondly, when an electron is emitted off a material, as in a photoemission process or a beta decay, can you show me how such electrons are treated as far as its description goes?
You mean a free electron? See any intro quantum textbook.

It appears that you are insisting that such an electron can be treated as a QM particle since it can be confined for a "very long time" inside such a device this size. Can you show me one specific example where such a treatment has been done and where this idea is applicable?
Why should I need to? Do you not believe solutions to the Schrodinger wave equation?