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Suppose that completely stopping a subatomic particle, such as an electron, could happen under certain conditions. What would be likely ways to get an electron to be perfectly still, or even just stop rotating the nucleus and collapse into it by electromagnetic forces? What would likely be required, below absolute-zero temperatures? Negative energy? Or could a 0 energy rest state not exist in any form, of any possible universe imaginable?

Let's say there was a magnetic field of a certain shape that we could postulate that is so intensely strong that if we put an electron in the center of it, it could not move at all in any direction. Would the energy requirement of the field be infinite? What would be the particle's recourse under this condition?

Further, suppose it were possible and one could trap an electron and stop all motion completely. What would this do to Heisenberg's Uncertainty Principle and/or Quantum Mechanics, because its position and momentum (0) would both be known? If it can be done, is Quantum Mechanics no longer an accurate model of reality under these conditions? Could we say QM is an accurate model under most conditions, except where it is possible to measure both position and momentum of a particle with zero uncertainty?

Clarification:

Please assume, confined in a thought experiment, that it IS possible to stop a particle so that is has 0 fixed energy. This may mean Quantum Mechanics is false, and it may also mean that under certain conditions the uncertainty commutation is 0. ASSUMING that it could physically be done, what would be likely to do it, and what would be the implications on the rest of physics?

Bonus Points

Now here's the step I'm really after - can anyone tell me why a model in which particles can be stopped is so obviously not the reality we live in? Consider the 'corrected' model is QM everywhere else (so all it's predictions hold in the 'normal' regions of the universe), but particles can be COMPLETELY stopped {{inside black holes, between supermagnetic fields, or insert other extremely difficult/rare conditions here}}. How do we know it's the case that because the uncertainty principle has lived up to testing on earth-accessable conditions, that it holds up under ALL conditions, everywhere, for all times?

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answer for your "bonus point": because hydrogen absorption lines are the same everywhere in the universe, only showing a net displacement due to the cosmological red-shift –  lurscher Aug 16 '12 at 6:17
    
The light from the inside of a black hole, or in regions outside our observable light cone both by definition do not reach us and are not detectable. I fail to see how hydrogen absorption lines = particles can never be stopped under any condition, at any location, ever. –  Ehryk Aug 16 '12 at 6:26
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6 Answers 6

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Your question is interesting, and gets specifically to the kinds of questions that quantum mechanics was intended to answer in the first place. It helps to understand the motivation behind the original Bohr model of the atom, and how that led to QM in the first place.

The problem Bohr was trying to address can be paraphrased as, "If an electron orbits a nucleus like a planet, why doesn't it gradually lose energy and spiral into the nucleus?" The answer came when Bohr realized that the orbital momentum was quantized, effectively meaning that since the electron had mass then by the relationship $p=mv$, the velocity was also quantized (note: this simple expression is more complicated when relativity is included, but the discussion can continue without including it). These quantized values of momentum/velocity are what one would call eigenvalues, or observables in quantum mechanics. Since the electron can only change orbits by given off specific quantities of energy instead of giving off energy continuously, it remained in a stable orbit relative to the nucleus, thus preventing it from spiraling in.

What is important to understand is the idea of the potential well. In the Bohr model, the electrons closest to the nucleus have higher velocity than the electrons further away. In other words, they have greater kinetic energy ($K.E. = \frac{1}{2} mass \times velocity^2$), but it had less potential energy since it was closer to the nucleus (obeying the relationship $P.E. = mass \times distance \times gravity$). However, the total energy ($K.E. + P.E.$) associated with orbits closer to the nucleus is less than those further away. So in order for the electron to move closer to the nucleus, energy must be given up. This is accomplished by the emission of a photon. Alternatively, if one wants to cause an electron to move into a more distant orbit, then one must add energy through use of a photon.

It is in contemplating how to determine the orbit of the electron that the uncertainty principle first became apparent. The only probe that we have available to determine the position of an electron in its orbit is a photon, and the photon must be of sufficient energy in order for it to be small enough to give a meaningful result, however if we use a photon small enough (in terms of wavelength), it will have enough energy to shift the electron into a different orbit, and then we would have to start the process all over again.

A free electron has sufficient energy to escape the nucleus. In other words, it has acquired sufficient energy to fill its potential energy deficit. If there were only one nucleus in the universe, the potential deficit would only be eliminated when the electron was at infinite distance from the nucleus. In that situation, the implication is that the electron would also have zero velocity. This situation is obviously unrealistic, first there are more than one nuclei in the universe, and second, to verify that a particle has zero velocity at infinite distance is clearly an impossible task.

If we move past the Bohr model and into more modern quantum mechanics, the question then is whether there are eigenstates that have eigenvalues for momentum of a particle that are equal to zero? It is important to review some basic facts about matrix operations and linear algebra

If 0 is an eigenvalue of a matrix A, then the equation A x = λ x = 0 x = 0 must have nonzero solutions, which are the eigenvectors associated with λ = 0. But if A is square and A x = 0 has nonzero solutions, then A must be singular, that is, det A must be 0. This observation establishes the following fact: Zero is an eigenvalue of a matrix if and only if the matrix is singular.

This means that the matrix in question is not of full rank. In QFT this has a very specific interpretation. The annihilation operator has the power to destroy the vacuum state and map it to zero. This situation is understood to be associated with the free field vacuum state with no particles. This state is necessary because it allows us to find vacuum state solutions for the associated quantum mechanical system.

By means of analogy, we can see that the solution to ground state problem is the solution to the homogeneous part of a differential equation.

The Schrodinger equation is a linear, homogeneous equation which governs evolution of the wave function of a particle. The solutions of the Schrodinger equation can be used to understand particle motion. The exact position and momentum of a particle can only be known if h (planck's constant) approaches zero, however, in quantum mechanics, planck's constant is fundamental to the theory, so this cannot occur in the single particle case. Because of this, momentum and position uncertainty establish an inverse relation to each other, and if the uncertainty of momentum is zero, then the uncertainty in position is infinite.

For these reason's it is not possible to talk about a particle "stopping" or being "stopped" in any meaningful or non-contrived sense.

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So what's stopping a ground state hydrogen electron from emitting that one last quanta as a photon and falling into the nucleus? –  Ehryk Sep 20 '12 at 16:30
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What stops an electron in hydrogen from falling and emitting a last photon? Conservation of energy: e+p->n+photon cannot proceed because the neutron is heavier than the electron+proton. Also the EM force does not allow that reaction even if the e has enough energy. However, the weak interactions allow e+p->n+neutrino, but again only if the e has enough energy to create a neutron from a proton. –  FrankH Sep 20 '12 at 16:49
    
@Ehryk A good question, and FrankH is correct. A good reference is to read the wikipedia article on electron capture en.wikipedia.org/wiki/Electron_capture –  Hal Swyers Sep 20 '12 at 17:42
    
Why can't it 'fall in' but not be captured by a proton? It seems through Quantum Tunnelling that an electron can sometimes be inside of a nucleus - so what's stopping it from just 'touching' the proton and sitting on its surface? –  Ehryk Sep 20 '12 at 18:48
    
First, quantum tunneling is possible because of the indefinite position/momentum of the particle. Since the particle's observed position is determined by the wave function, when there is a potential barrier, the solution slightly bleeds past the classically defined barrier. The proton, like the electron, do not possess a classical "surface". With the advent of QFT we understand the electron and proton as regions of space with intense particle-antiparticle creation/annihilation events where there are net positive amounts of the properties associated with a particle. –  Hal Swyers Sep 21 '12 at 1:20
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This is just a misunderstanding--- "no motion" in quantum mechanics is a different concept than "no motion" in classical mechanics. At zero temperature, nothing stops. Spherical uncharged black holes don't stop particles at the singularity, they absorb particles and time just ends at the singularity for the infalling matter. The wavefunctions are not made to stop.

You can stop an electron by putting it in the ground state of Hydrogen. This is what it means to be stopped in quantum mechanics.

The reason one can be sure that the uncertainty principle applies to more than what we have seen is that it is impossible to make part of the world classical and part quantum mechanical, as understood by Bohr and Rosenfeld in the early days of quantum mechanics. The fact that the electron has an uncertainty principle means that there would be a contradiction if something else did not, because this would allow you to violate the uncertainty principle for electrons by interacting them with this new thing.

If the world is classical underneath, the classical variables will have very little relation to the position and momentum of classical point particles. This question is annoying, and it does not deserve any more attention than what it has gotten.

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You started with a good line of reasoning, then you justified that everything must obey the uncertainty principle because we observe (so far) that electrons obey it which is a bit of circular logic. Then you end up criticizing the question, which is the exact prejudice that made me ask it in the first place. –  Ehryk Sep 14 '12 at 21:42
    
It's tolerable to say 'the non-commutation of wave-forms is an assumption of QM, and we need that for QM to work', and continue with the theory. It's then great to move to 'IF particles can be represented as wave-forms always and everywhere, at every time, under all conditions, QM is likely true.' It's not then okay to jump to 'therefore our assumption for the theory is unquestionable and shouldn't even be discussed.' The way this is treated seems completely indistinguishable to me from a faith/god claim - just assume the X, therefore it is true. Then rationalize God after the fact using X. –  Ehryk Sep 14 '12 at 21:47
    
"There can never be a completely arrested particle, under any conditions, ever, at any time and under any circumstance" is an extraordinary claim, and the evidence for it is surprisingly vapid. ("We use that assumption to make a theory, and the theory's accurate from our limited corner of the universe with limited means of observation and measurement"). –  Ehryk Sep 14 '12 at 21:54
    
Please see my other question, if you haven't already, though perhaps that will just annoy you as well. –  Ehryk Sep 14 '12 at 22:31
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@Ehryk: The annoying thing is that, yes, quantum mechanics could be wrong, but no, not in this obvious classical way. It might be wrong in a subtler holographic way. This is something people like 'tHooft investigate, and it is not the same thing as just saying "oh, the electron has a secret position and momentum that we just don't know", because this idea is demonstrably false. –  Ron Maimon Sep 15 '12 at 0:44
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There exists a huge number of experimental evidence that in the subatomic world nature is using quantum mechanics. In quantum mechanics bound states always have the first energy level above 0 energy. Your magnetic field thought experiment creates a bound stated in the collective potential.

Free states have to obey the HUP and therefore cannot have 0 fixed energy.

Your question is another form of trying to impose the classical thinking of macroscopic physics to the microcosm; it cannot be done because it has not been done after numerous experiments.

Edit An analogy: Take the ocean and a tiny volume dV on the surface of it. This volume bobs up and down depending on the wave activity, and there is always some wave activity in the ocean. Does it make sense to ask "under what circumstances will this dV be at rest with respect to the earth?" "Could one use special waves to keep this specific dV motionless?".

The subatomic particles are probability waves and at that level one has as little control as in the ocean picture as far as localization goes, because any fields one can use at the subatomic level are also probability waves. Now we happen to have a solid theory with QM which can predict for us mathematically that there will always be some energy associated with a subatomic particle that will not allow it to be at rest, so we do not need to handwave, as with the ocean where we can only use statistical arguments.

Edit2

Ehryk : A thought experiment in physics has to start with the known physics data and theories and extend it to uncharted territory. What everybody is telling you is that the known physics theories that have fitted perfectly all known experimental data, tell us that even in a thought experiment using known theories there cannot be a "stopped" particle in the microcosm.

Of course there can be science fiction physics, i.e. where one assumes new laws of physics. Then the burden is on the one who suggests these new laws to suggest experiments that will validate them. To imagine situations that have not been observed ever and are not in the realm of accepted theoretical calculations means science fiction.

Now if there were an experiment with solid data that demonstrated a "stopped particle" or "stopped particles" then every physicist would look up and scramble to try and understand it.

This is a physics board, and ultimately physics is about experimental data fitted by theories, not random models of how natural laws could be.

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It cannot ever be done (ever, by anyone or anything) because we haven't done it yet after some experimentation? –  Ehryk Aug 16 '12 at 5:11
    
Because the quantum mechanical nature of the microcosm does not allow it, and this is fully consistent with experiments. –  anna v Aug 16 '12 at 5:19
    
@Ehryk, when a physicist says "it cannot be done" it actually means "assuming its true everything we know about physics, it cannot be done". As long as the laws of physics don't change or some unknown deeper law are unhidden, then it will not happen. –  lurscher Aug 16 '12 at 5:20
    
Why cannot there be a subtle improvement of QM under certain conditions, the same way Newtonian Physics was 'almost there' until we observed relativistic effects? What IF a particle could be stopped? QM holds for most cases, except <insert certain conditions here>? –  Ehryk Aug 16 '12 at 5:27
    
There are theorists working on such deep matters. Have a look at a recent post by 'tHooft on this forum: physics.stackexchange.com/questions/30065/… . At the moment they are not proposing any experiments that would refute QM. They are attempting to get QM out of hidden variables, but this will not change the agreement of QM with the subatomic world data. –  anna v Aug 16 '12 at 5:34
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you have provided your own answer: as long as the uncertainty conmutation relations hold (and it does hold, have no doubt about it), there is no going to be such a thing as standing perfectly still for particles.

But wait, there is the quantum zeno effect, which does a similar thing, but not quite the same as you ask; it keeps the quantum state from evolving.

See a related question: Quantum Zeno effect and unstable particles

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I have doubts. What if the uncertainty commutation did not hold? Why can't it be given up, even for an alternate model that is not the reality we live in (like hyperbolic geometry, for example)? –  Ehryk Aug 16 '12 at 5:13
    
sure, you can do such things, but then its not real physics anymore. it becomes a mathematical exercise. Now, ocasionally such mathematical exercises end up having great unexpected connections with real physics, as the case with twistor theory. But for each twistor theory there are thousands of dead ends –  lurscher Aug 16 '12 at 5:14
    
The part I'm missing is why uncertainty isn't questioned, or why there's never a "maybe the commutations don't always hold" discussion. Whenever something conflicts with it, the next statement is <therefore that something is wrong>, as though it's set in stone and its not worth questioning. How do you KNOW there aren't certain conditions (that perhaps we haven't the technology or ability to create) in which the uncertainty commutations do not hold? –  Ehryk Aug 16 '12 at 5:20
    
(in case I didn't state my question well enough, I did not provide my own answer. I want to know: what if the uncertainty commutation does not hold under certain conditions? Not: it holds, don't doubt it.) –  Ehryk Aug 16 '12 at 5:23
    
the ultimate answer is about resources; how much time do we allocate to research what? and for what we need educated guesses. When observations have hundreds or thousands of experiments backing them, people tend to invest exponentially less resources to try to debunk them. –  lurscher Aug 16 '12 at 5:23
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For a long time people have been interested in ultra low temperatures (as far as I know the current reccord is 0.1 nanokelvin). In this range of temperature, QM is all but desappearing. On the contrary, we observe long range QM effects, all observations tends to show that QM waves strech up and overlap. The most popular example is Bose Einstein condensates but it has also been shown that ultra-cold gas can chemically react at distances up to 100 times greater than they can at room temperature (Science 12 February 2010: Vol. 327 no. 5967 pp. 853-857 )

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Is absolute zero equivalent to motionless particles/electrons? –  Ehryk Aug 16 '12 at 7:38
    
yes. 3/2*k*T=E. If T goes to zero, so it is for the energy E. –  Shaktyai Aug 16 '12 at 14:09
    
So if there was an atom of Hydrogen that was AT absolute zero, exactly, the electron would STOP rotating, and be pulled into the nucleus? And it's position and momentum could both be known at the same time? –  Ehryk Aug 16 '12 at 18:35
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The lower the temperature the more delocalized the particle is, so at T=0 K the electron and proton both occupy the whole space... delta(x) goes to infinity and delta (p) to zero. –  Shaktyai Aug 16 '12 at 21:34
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The uncertainty principles is not QM tied. It is a relation between a function and its Fourier transform. If you want to get rid of it, you need to get rid of p being the fourier transform of x. I agree with you we can imagine anything we like, but don't ask for clues based on what we currently know to shoulder your dreams. –  Shaktyai Aug 17 '12 at 11:20
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It is quite easy to make electron to stop (that is make it having zero impulse and kinetic energy). But once it has zero impulse, it is smoothed over a volume. You can see it as having big dimensions rather than being point-like.

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