Regarding the Quantum Mechanics of Electrons

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Anthyo
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Regarding the Quantum Mechanics of Electrons

Postby Anthyo » Fri Nov 05, 2010 9:20 pm UTC

As a preface to the question proper, I'd have you know that I am a high school physics student. I would approximate my level of knowledge as that of a student who has finished his first year of college physics courses. My knowledge of anything beyond Newtonian mechanics and basic electromagnetism is sketchy at best.

Regarding the electron cloud of an atom - I am told that electrons move about an atom randomly. That is, they do not obey any direct observable laws of motion. Rather, each electron has a certain zone about which it moves willy-nilly and it may be anywhere in that zone at any given time. Due to the Heisenberg uncertainty principle (is this a product of human limitations or is it bona fide unknowable?)we cannot know the location and velocity of an electron simultaneously. My questions are thus:

Are electrons still subject to normal laws of motion? That is, are electrons' motion still describable in terms of electrostatic forces and gravity despite the fact that they move so fast we cannot observe these interactions? Could their motion be predicted in theory? Would, for example, Laplace's Demon be able to tell us where an electron was and where it was headed based on its precise knowledge and application of such forces? Or is there something entirely different going on here?
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Re: Regarding the Quantum Mechanics of Electrons

Postby Tass » Fri Nov 05, 2010 10:34 pm UTC

Yes, electrons are subject to the law called the Scrödinger equation. It doesn't move willy-nilly rather it is everywhere at once, in a way. It is more some places than others in a way governed by the mentioned equation.

It is really hard to explain properly without giving a complete university level course in introductory quantum mechanics.

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Re: Regarding the Quantum Mechanics of Electrons

Postby GeorgeH » Fri Nov 05, 2010 10:53 pm UTC

^What Tass said. Also, because I already typed it:

Anthyo wrote:Are electrons still subject to normal laws of motion?

Basically. The Schrödinger equation from which all that is derived is basically a complex statement ( :wink: ) that E=T+U (total energy = kinetic energy + potential energy.)

There are many ways to interpret QM, but you need to be thinking of the electron less as a little ball shooting around in space and more as a probability distribution. In other words those orbital clouds don't describe where a little ball might be, they directly describe what is actually there. There isn't a god or a demon that can tell you where the little ball is, because the little ball doesn't exist; it's nothing but an occasionally useful fiction (note that this is only kind of true, but without more background in QM it's hard to explain properly.)

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Re: Regarding the Quantum Mechanics of Electrons

Postby Aiwendil42 » Fri Nov 05, 2010 11:13 pm UTC

the Heisenberg uncertainty principle (is this a product of human limitations or is it bona fide unknowable?)


The latter. This has to do with the properties of the wave functions that are associated with particles in QM. Basically, the wave function of a particle in terms of momentum is the Fourier transform of its wave function in terms of position. If an electron had a precise position {x,y,z}, this would mean its wave function in position had the value zero everywhere except at the point {x,y,z} (it would be what's called a "delta function"). But the Fourier transform of a delta function is a sinusoidal function (actually a sinusoidal function with both real and imaginary parts), which of course continues off to infinity in both directions. So if a particle could be completely localized in position, it would have no well-defined momentum at all (I guess you could say it would have every possible momentum at once) and vice versa. In fact, it's generally true that the more localized you make a function, the less localized its Fourier transform will be.

That may not make any sense if you haven't heard of Fourier transforms and wave functions, but the upshot is that the Uncertainty Principle arises from the mathematical properties of the objects in quantum mechanics, not from limitations specifically inherent in measurement, human or otherwise.

Are electrons still subject to normal laws of motion?


Tass and GeorgeH have mentioned the Schrodinger equation, which is the "law of motion" for electrons in QM. I would only add that two things are worth emphasizing. First, to directly answer your question, no, electrons do not obey the "normal laws of motion", if what you mean by that is the laws of Newtonian physics. In other words, Schrodinger's equation and Newton's laws assert different things; they contradict each other, and if you were to perform sufficiently sensitive tests, you'd find that the data agree much better with Schrodinger's equation than with Newton's. Second, this doesn't just apply to electrons - it applies to everything: protons, atoms, molecules, cars, people, galaxies, etc. All these things follow (well, nearly follow) Schrodinger's equation and violate Newtonian mechanics. The thing is, for sufficiently large things, the results Schrodinger's equation gives you will be really, really, really close to the results Newton gives you. That's why Newtonian mechanics is still used - if you're dealing with macroscopic things, it's more than good enough. It's only when you deal with very small things that you notice that Newton's equations aren't right, and QM is much better.

Actually, if you deal with things that are also very, very fast you find that Schrodinger's equation is wrong too, and you need what are called Quantum Field Theories. You might have heard of the Standard Model of particle physics - this is a Quantum Field Theory, currently the most accurate one we have. But even this turns out not to be quite true, because it doesn't offer a quantum description of gravity. The "Theory of Everything", as the media calls it, is the conjectured theory that will incorporate gravity and supercede the Standard Model, just as the Standard Model superceded Quantum Mechanics and Quantum Mechanics superceded Newtonian Mechanics.
Last edited by Aiwendil42 on Fri Nov 05, 2010 11:36 pm UTC, edited 1 time in total.

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Re: Regarding the Quantum Mechanics of Electrons

Postby Dopefish » Fri Nov 05, 2010 11:24 pm UTC

Due to the Heisenberg uncertainty principle (is this a product of human limitations or is it bona fide unknowable?


Bona fide unknowable.

Given the appropriate tools (theoretical or experimental), we can figure out quite precisely what the uncertainty in various quantities are, but we can't overcome those uncertainties (there are essentially an infinite number of uncertainty relations, the Heisenberg one relating postion and momentum is just the most well known) , because they're fundementally there. That is, a particle is in a super position of states, with the weighting of each state being related to the probability of observing it. It's not a matter of it being in one particular state and we just don't know what it is, it really is in multiple states at once, and doesn't take on a particular value until we observe it.

Einstein wasn't comfortable with that (and most people aren't, but you eventually learn to accept it), but various experiments have been done to show that a particle is in a super position of several states, rather than just one state. It goes against all our classical intuition, but without QM, there are a number of phenomena that simply can't be explained using classical methods, so it is something to be accepted as true (/learned about when you get the oppurtunity).

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Re: Regarding the Quantum Mechanics of Electrons

Postby BlackSails » Sat Nov 06, 2010 1:30 am UTC

Anthyo wrote:Regarding the electron cloud of an atom - I am told that electrons move about an atom randomly. That is, they do not obey any direct observable laws of motion. Rather, each electron has a certain zone about which it moves willy-nilly and it may be anywhere in that zone at any given time.



The electron isnt moving around the nucleus. If it were, according to maxwell's laws, it would have to radiate, which would mean it loses energy and eventually drops into the nucleus. The electron is in a stationary state. This means that the quantum mechanical state the electron is in does not evolve with time (barring external influences). It isnt at any particular point around the nucleus, but you cant quite say it is at all of them.

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Re: Regarding the Quantum Mechanics of Electrons

Postby Sagekilla » Sat Nov 06, 2010 4:48 am UTC

There was a post on this a couple of days ago. I'll have to go and dig it up, but to summarize the key points:
(Edit: when I first wrote this post it was supposed to be a summary, it ended up being a bit longer)

1) The Schrödinger equation describes essentially the evolution of an ensemble of identical particles.
This just means to say that if you had a huge number of identical and independent particles, their time evolution
would be described by the Schrödinger equation.

2) When you solve the Schrödinger equation, you're solving for a probability function. This describes where the particle
is most likely to be. When one says that the electron is "smeared" in a cloud around a nucleus, we're actually saying that
the probability distribution gives us a range of locations where it could possibly be (the cloud).

3) As was mentioned before, one consequence is that we can't know the position and momentum (which are conjugate variables)
precisely at the same time. Since to get from real space (position) to momentum space you need to take a Fourier transform,
we are inherently mathematically by the uncertainty principle.

4) Electrons have quantized angular momentum, which leads to quantized radii which they orbit the nucleus. In other words,
you'll only find the electron at a certain radius from the nucleus. However, an electron's probability function forms a standing
wave when you describe how the electron orbits the nucleus. This just means that the electron doesn't give off radiation as it
orbits the nucleus, as a classical treatment would suggest. However, it can give off radiation in the form of photons when it
falls from an excited state to the ground state (in other words, when it jumps down to one of the orbits that's closer to the nucleus).

5) The ground state doesn't have zero energy as is common in classical mechanics. Here, it has some finite and quantized energy.
Also, when you add electrons to a nucleus, they can't all occupy the ground state (the nearest orbital). Instead, the progressively fill
up shells, which themselves are only at specific radii. This is because electrons are what are known as fermions, and obey the
Fermi-Dirac statistics. Because of this, you can't have two electrons occupying the same state. This part gets a little bit more involved
since it's related to solving the Schrödinger equation, but you basically can't have two particles with the same set of quantum numbers,
also known as the state.

If I've made any mistakes, to those of you more knowledgeable than me let me know and I'll make the appropriate changes :)

To any moderators: Perhaps we should have a "Commonly asked questions about Quantum Mechanics," just like the one for GR?
If this is already addressed in that thread, then forgive my ignorance and simply forget about this little tidbit.
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Re: Regarding the Quantum Mechanics of Electrons

Postby Sir_Elderberry » Sat Nov 06, 2010 4:56 pm UTC

The relativity thread was originally just "common questions"--relativity's just the most common, so we put it in the thread title.
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Re: Regarding the Quantum Mechanics of Electrons

Postby Wnderer » Sat Nov 06, 2010 5:27 pm UTC

2) When you solve the Schrödinger equation, you're solving for a probability function. This describes where the particle
is most likely to be. When one says that the electron is "smeared" in a cloud around a nucleus, we're actually saying that
the probability distribution gives us a range of locations where it could possibly be (the cloud).


The electron is smeared out. The electron is in all those positions at once. The electron is a wave. The probability isn't in the electrons position but in a particle type event occurring at that position. Look at the emission by emission double slit experiment. http://en.wikipedia.org/wiki/Double-slit_experiment The electron passes through both slits simultaneously and interferes with itself. It then hits the screen and then there is probability of a particle type interaction at the screen. Throw enough particles at the screen and the probability distribution becomes apparent, just like rolling a die enough times shows it 1/6th distribution for its sides. The electron is a particle during particle type events and it is a wave during wave type events.

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Re: Regarding the Quantum Mechanics of Electrons

Postby BlackSails » Sat Nov 06, 2010 5:36 pm UTC

Wnderer wrote:
The electron is smeared out. The electron is in all those positions at once.


That cant be right, because otherwise the electron would be an extended distribution of charge, and would have different properties from what we observe.

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Re: Regarding the Quantum Mechanics of Electrons

Postby Wnderer » Sat Nov 06, 2010 5:57 pm UTC

BlackSails wrote:
Wnderer wrote:
The electron is smeared out. The electron is in all those positions at once.


That cant be right, because otherwise the electron would be an extended distribution of charge, and would have different properties from what we observe.

Interesting. Do you have a link? I see in that slit experiment, a single electron passing through both slits at the same time as a wave. The charge distribution you are referring to may be being measured through particle type events, where the electron now has single position. For me the question from these experiments is not what is the exact nature of the electron, but what is the difference between a particle type event and a wave type event. Why isn't passing through a slit a particle type event?

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Re: Regarding the Quantum Mechanics of Electrons

Postby BlackSails » Sat Nov 06, 2010 6:19 pm UTC

I believe it was in Feynman's QM (or stat mech) textbook, but I read through a lot of textbooks last year, and it might be somewhere completely different.

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Re: Regarding the Quantum Mechanics of Electrons

Postby thoughtfully » Sat Nov 06, 2010 7:10 pm UTC

Not only that, but it doesn't jibe with the version of the experiment that only fires one eletron/photon/etc art a time, unless its interfering with its past/future selves, which wouldn't be the craziest thing about QM, actually.
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Re: Regarding the Quantum Mechanics of Electrons

Postby Wnderer » Sat Nov 06, 2010 9:45 pm UTC

thoughtfully wrote:Not only that, but it doesn't jibe with the version of the experiment that only fires one eletron/photon/etc art a time, unless its interfering with its past/future selves, which wouldn't be the craziest thing about QM, actually.


That is exactly the case I'm talking about.

from http://en.wikipedia.org/wiki/Double-slit_experiment

When electrons are fired singly through a double-slit apparatus they do not cluster around two single points directly on lines between the emitter and the two slits, but instead one by one they create an interference pattern. However, they do not arrive at the screen in any predictable order. In other words, knowing where all the previous electrons appeared on the screen and in what order tells us nothing about where any future electron will hit (although we can calculate the probability of it striking at any specified point).[34]


The electron passes simultaneously through both slits and interferes with itself just like a wave but then it hits the film and acts like a single particle. Why? Shouldn't it have a probability of passing through one slit or the other just like it has a probability of hitting one location or another on the screen? It has something to do with the information it leaves behind. When it passes through the slits it interferes and leave information about its wavelength and therefore its momentum so it can't have a position. When it hits the film it leaves information about its position so it loses all information about momentum. The part in the wiki that's new to me, is they can now vary the proportions of positional information vs wavelength information they get for the particles passage through the slit. Wild

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Re: Regarding the Quantum Mechanics of Electrons

Postby BlackSails » Sat Nov 06, 2010 10:10 pm UTC

If you want something bizzare, look up the Wheeler delayed choice quantum eraser.

Also the elitzur vaidmann bomb inferometer.

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Re: Regarding the Quantum Mechanics of Electrons

Postby thoughtfully » Sat Nov 06, 2010 11:12 pm UTC

BlackSails wrote:If you want something bizzare, look up the Wheeler delayed choice quantum eraser.

Also the elitzur vaidmann bomb inferometer.

Yeah, because EPR/Bell inequality/Aspect experiments are soo mundane nowadays :twisted:
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Re: Regarding the Quantum Mechanics of Electrons

Postby Aiwendil42 » Sun Nov 07, 2010 1:04 am UTC

The electron passes simultaneously through both slits and interferes with itself just like a wave but then it hits the film and acts like a single particle. Why? Shouldn't it have a probability of passing through one slit or the other just like it has a probability of hitting one location or another on the screen?


The usual way to talk about this is to say that when the electron hits the screen, the wave function collapses. This "collapse of the wave function" is the heart of the so-called "measurement problem" of QM.

Essentially, QM is a strange theory in the following way. Most of the time, wave functions evolve in time according to the Schrodinger equation. However, when a measurement is made, the wave function for an instant stops obeying the Schrodinger equation and instead collapses into an eigenstate of the value being measured. It then immediately begins evolving in accordance with Schrodinger's equation again. It's worth stressing how unusual this is in comparison with classical theories. In classical mechanics, Newton's laws are always true; there is a single set of rules that describes the evolution of its objects for all time. In QM, there are two fundamentally different sets of rules: the linear, deterministic dynamics of the Schrodinger equation and the non-linear, probabilistic Born rule for the collapse of the wave function; and one rule applies at certain times, the other at other times.

This by itself is strange but would not be problematic if the theory stated unambiguously under what conditions each of the rules applies. However, the standard formulation of QM does not provide such an unambiguous statement; it says that the collapse occurs when a "measurement" is made, but of course "measurement" is a vague term. This is the "measurement" problem, and the "interpretations" of QM that you hear about are essentially different proposals for the solution of this problem. Some attempt to provide a more precise condition for collapse; some argue that it doesn't matter when collapse occurs, just that it does; some do away with the collapse entirely.

Each interpretation would probably answer your question (about why particle passes through both slits but then chooses just one location on the screen) differently. For instance, Wigner would tell you that the particle doesn't actually choose one location (i.e. the wave function doesn't collapse) when it hits the screen, but only when a conscious observer looks at the screen. The GRW interpretation would tell you that the wave function collapses when it hits the screen because it has interacted and become entangled with many other wave functions, and one of those wave functions has collapsed randomly. Everett's disciples would tell you that the wave function actually never collapsed at all. Most working physicists would probably tell you "Come on, in practice you and I both know very well what it means to make a measurement, so just shut up and calculate."

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Re: Regarding the Quantum Mechanics of Electrons

Postby Tass » Sun Nov 07, 2010 10:32 am UTC

Then there is of course us who says it never collapses, but rather continues to evolve unitarily according to the Schrödinger equation. When you observe a collapse it is just because you have become entangled with the system.

I find that this is the most simple, elegant and most likely true explanation. However, it has the implication that what we observe today is only a tiny part of the universes wave-function and that other parts would be observed by other you's. Many people don't like this and therefore tries to explain away the conclusion, but I say nature does not care what you like.

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Re: Regarding the Quantum Mechanics of Electrons

Postby mr-mitch » Sun Nov 07, 2010 1:52 pm UTC

Tass wrote:Then there is of course us who says it never collapses, but rather continues to evolve unitarily according to the Schrödinger equation. When you observe a collapse it is just because you have become entangled with the system.

I find that this is the most simple, elegant and most likely true explanation. However, it has the implication that what we observe today is only a tiny part of the universes wave-function and that other parts would be observed by other you's. Many people don't like this and therefore tries to explain away the conclusion, but I say nature does not care what you like.


What do you mean by become entangled with the system?

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Re: Regarding the Quantum Mechanics of Electrons

Postby Aiwendil42 » Sun Nov 07, 2010 7:00 pm UTC

Then there is of course us who says it never collapses, but rather continues to evolve unitarily according to the Schrödinger equation. When you observe a collapse it is just because you have become entangled with the system.


Yes, this is what I meant when I mentioned Everett's ideas - although actually there are various flavours of Everett-style interpretations, from the (in)famous "many worlds" through variations such as "many histories" and "many minds" to the somewhat baffling "bare theory". As it happens, my own views are more or less along these lines, though with some logical positivist caveats to eliminate what I see as meaningless metaphysical statements. Actually, I think that when you strip away the pseudo-statements, all (meaningful) interpretations without collapse are equivalent (even non-Everett types such as Bohm's interpretation).

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Re: Regarding the Quantum Mechanics of Electrons

Postby Sagekilla » Wed Nov 10, 2010 4:18 pm UTC

Wnderer wrote:
2) When you solve the Schrödinger equation, you're solving for a probability function. This describes where the particle
is most likely to be. When one says that the electron is "smeared" in a cloud around a nucleus, we're actually saying that
the probability distribution gives us a range of locations where it could possibly be (the cloud).


The probability isn't in the electrons position but in a particle type event occurring at that position. Look at the emission by emission double slit experiment. http://en.wikipedia.org/wiki/Double-slit_experiment The electron passes through both slits simultaneously and interferes with itself. It then hits the screen and then there is probability of a particle type interaction at the screen. Throw enough particles at the screen and the probability distribution becomes apparent, just like rolling a die enough times shows it 1/6th distribution for its sides. The electron is a particle during particle type events and it is a wave during wave type events.


Just as an aside, what you're saying here is exactly what I said above. The wave function (well the squared wave function)
describes where the electron is most likely to be. It doesn't out right say "Here it is!" Just where it most likely will be.

The double slit is nothing contrary to what I said either. Classically, as a particle, you would expect the electron to go through
one slight or the other. Like you said, the wave function is just interfering with itself and producing a different probability
distribution of where it would be.
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Re: Regarding the Quantum Mechanics of Electrons

Postby SU3SU2U1 » Wed Nov 10, 2010 5:38 pm UTC

Tass wrote:Then there is of course us who says it never collapses, but rather continues to evolve unitarily according to the Schrödinger equation. When you observe a collapse it is just because you have become entangled with the system.

I find that this is the most simple, elegant and most likely true explanation. However, it has the implication that what we observe today is only a tiny part of the universes wave-function and that other parts would be observed by other you's. Many people don't like this and therefore tries to explain away the conclusion, but I say nature does not care what you like.


This has its own host of problems- on an aesthetic level, why are we "aware" of only one of us? If I'm actually in a massive super-position with others, why don't I know? Isn't this elevating consciousness ala Wigner?

On a technical level, the various proofs that you recover the standard Born rule for probabilities if you create large ensembles only actually works out if you have infinite degrees of freedom. For finite degrees of freedom, observers in most branches will not recover the born rule.

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Re: Regarding the Quantum Mechanics of Electrons

Postby Moose Hole » Wed Nov 10, 2010 6:05 pm UTC

One way to think of it is that we interact with the world at a high level. The elementary particles around us come together to create particles of sand, sound and light waves, and things like that which we can readily observe. We describe elementary particles as particles or waves because we are familiar with how those things behave. But when you take those types of things apart enough to get to the elementary particles, they don't really behave in any way we're used to.

If you had a tire and you didn't understand it, I could tell you that sometimes it acts like a car, and other times like a bouncy doughnut. You understand what cars and bouncy doughnuts are because you are familiar with them, but the tire is really neither of those.


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