I'm just wondering, what causes the existence of quantum levels?

[d-wat]

i'm new here and I'm not sure if this has been asked before, but I'm just wondering about the cause of its existence. I'm speculating that it may be caused by interaction between a proton and electron together with their masses, if so, how?

 

[madhatter106]

I'm going to help you out a bit here and say that most likely you're not going to get an answer as it's not a question. I'll use a phrase I do actually use with my wife when she asks me things that are not questions. it's like asking
"Why does July taste like the color purple?"
It's not said to be rude or mean but to bring humor in asking a non-question.

your question contains asking why things are the size they are, is that your question?

 

[LostConjugate]

Boundary conditions..

The blackbody radiation experiment had boundary conditions.

QM equations have boundary conditions, even if we have to make the length between boundaries go to infinity.

Let's remove boundaries then.

Take the free particle hamiltonian energy eignvalue equation. The energy eigenvalues are p^2/2m where p is continuous. No more quantum levels.

 

[madhatter106]

Boundary conditions..

The blackbody radiation experiment had boundary conditions.

QM equations have boundary conditions, even if we have to make the length between boundaries go to infinity.

Let's remove boundaries then.

Take the free particle hamiltonian energy eignvalue equation. The energy eigenvalues are p^2/2m where p is continuous. No more quantum levels.

Okay but that's not the cause for that scale, it's the equations to work in those scales. And unless I misunderstand your hamiltonian Eq, your setting the energy level of the plane wave high enough so that the eigenvalues becomes close enough to be free particles?, that really doesn't do away with the quantum level but removes that upper boundry. do correct me if I'm off on that.

To me the question from the OP is the equivalent of asking what is the cause for finer resolution? which really isn't a question at all.

 

[d-wat]

I'm going to help you out a bit here and say that most likely you're not going to get an answer as it's not a question. I'll use a phrase I do actually use with my wife when she asks me things that are not questions. it's like asking
"Why does July taste like the color purple?"
It's not said to be rude or mean but to bring humor in asking a non-question.

your question contains asking why things are the size they are, is that your question?

i don't think it is a non-question at all, i may just be referring to reality as to what might be a physical or at least a theory that may explain its physical cause, but just to make my self clearer, i will rephrase my question: what I'm asking is what causes the existence of the the thing that keeps electrons in place?

 

[LostConjugate]

what I'm asking is what causes the existence of the the thing that keeps electrons in place?

The Uncertainty Principle. As the electron becomes more isolated in space (closer to the nucleus) the electron's probability for a higher and higher momentum increases. This is another example of the wave like properties of an electron, uncertainty has always been a factor in even classical wave mechanics.

The boundary conditions of the central potential determine exactly what the lowest state can be.

Funny how uncertainty gives us certainty in so many things, in the end its uncertainly certain.

 

[ansgar]

The Uncertainty Principle. As the electron becomes more isolated in space (closer to the nucleus) the electron's probability for a higher and higher momentum increases. This is another example of the wave like properties of an electron, uncertainty has always been a factor in even classical wave mechanics.

The boundary conditions of the central potential determine exactly what the lowest state can be.

Funny how uncertainty gives us certainty in so many things, in the end its uncertainly certain.

stop miss using the uncertainty principle! it has to do with observations not energy levels

Have you done your QM class boy??...

 

[LostConjugate]

stop miss using the uncertainty principle! it has to do with observations not energy levels

Have you done your QM class boy??...

Huh?

http://www.physics.sfsu.edu/~greensit/book.pdf

Page 90 "Why the hydrogen atom is stable"

 

[SpectraCat]

stop miss using the uncertainty principle! it has to do with observations not energy levels

Have you done your QM class boy??...

You might want to watch your tone ... especially since you are wrong. The HUP has to do with whether or not observable quantities can be simultaneously well-defined ... it is completely independent of measurement or observation. The HUP sets a lower limit on the width of distributions from which non-commuting observables are sampled, so it is often explained in terms of observed distributions for repeated measurements. However it is far more fundamental than that, and actually is directly derivable from the postulate that quantum states are representable as vectors in a Hilbert space.

 

[ansgar]

Huh?

http://www.physics.sfsu.edu/~greensit/book.pdf

Page 90 "Why the hydrogen atom is stable"

those derivations are just heuristic pseudo arguments found in introductory books

 

[ansgar]

You might want to watch your tone ... especially since you are wrong. The HUP has to do with whether or not observable quantities can be simultaneously well-defined ... it is completely independent of measurement or observation. The HUP sets a lower limit on the width of distributions from which non-commuting observables are sampled, so it is often explained in terms of observed distributions for repeated measurements. However it is far more fundamental than that, and actually is directly derivable from the postulate that quantum states are representable as vectors in a Hilbert space.

are you cute?

I said that HUP has to do with measurements and not energy levels, I had no time to give a full derivation of HUP

 

[LostConjugate]

Enlighten us ansgar.

 

[humanino]

those derivations are just heuristic pseudo arguments found in introductory books

Yes, but if you questioned them seriously, you would be aware that those derivations can be carried over quite far in full QFT, and although they must certainly be taken with a grain of salt, still they remain quite suggestive.

In "The quantum vacuum, introduction to QED" Milonni discusses in section 2.6 how the momentum-position commutator for a charged particle would be damped exponentially to zero without fluctuations of the electromagnetic field to which it couples. That is because we can couple a charged particle to its own field.

What we have here is an example of a "fluctuation-dissipation relation." Generally speaking, if a system is coupled to a "bath" that can take energy from the system in an effectively irreversible way, then the bath must also cause fluctuations. The fluctuations and the dissipation go hand in hand; we cannot have one without the other. In the present example the coupling of a dipole oscillator to the electromagnetic field has a dissipative component, in the form of radiation reaction, and a fluctuation component, in the form of the zero-point (vacuum) field; given the existence of radiation reaction, the vacuum field must also exist in order to preserve the canonical commutation rule and all it entails.

Then in section 3.3 he goes on to atomic stability. He applies the same ideas, balancing how much the electron absorbs energy from the vacuum to how much it gives away, and obtains... Bohr quantization condition

This "derivation" of the Bohr quantization condition obviously should not be taken very seriously. It suggests only how Bohr's quantization condition, at least for n = 1, might have been interpreted by physicists in 1913. We now know that the vacuum field is in fact formally necessary for the stability of atoms in quantum theory: as we saw in Section 2.6, radiation reaction will cause canonical commutators like [x,p] to decay to zero unless the fluctuating vacuum field is included, in which case commutators are consistently preserved.

 

[humanino]

Enlighten us ansgar.

"Uncertainty" is quite a badly chosen name. The inequality is nothing but the Cauchy-Schwarz inequality. This is so well-known, it is in the introduction of a wikipedia article. Seriously, this is elementary Hilbert space theory.

 

[LostConjugate]

It is because of this inequality and functions having the properties of vectors that we have uncertainties right?

 

[humanino]

It is because of this inequality and functions having the properties of vectors that we have uncertainties right?

There is no "uncertainty" per se in the inequality, since there are some people advocating non-local hidden variable theories who will interpret that all variables are always well-defined, and they have not convincingly been proven wrong.

Second, there is no "principle" per se in the inequality either, as explained for instance here

The reason why the uncertainty theorem shouldn't be called a "principle" is that a principle is an idea, usually stated in non-mathematical terms, that you can use to find an appropriate mathematical structure for a new theory that you're trying to find. The "principle" restricts the number of mathematical structures you can use, because you're only looking for theories in which a mathematical statement that resembles the principle can be derived as a theorem.

The "HUP", or rather the statement that should be called the HUP, is a statement that predates QM. It was used to find QM. The inequality that people insist on calling "the HUP" is a theorem derived from the axioms of QM.

Other examples of "principles" in physics are "Einstein's postulates" (which are even more inappropriately named than "the HUP", because "postulate" is a synonym for "axiom", and these aren't even mathematical statements) and "the equivalence principle". The former can help you guess that Minkowski spacetime is an appropriate model of space and time, and the latter can help you guess that some other 4-dimensional smooth manifold with a Lorentzian metric determined by a bunch of fields on that manifold, might be an even better choice.

It bugs me a bit every time I see someone refer to the HUP, Einstein's postulates or the equivalence principle when they want to prove something. It sounds like they're referring to the ideas that led to the discovery of the theories, when they could and should be using the actual theories. But most of the time, what they have in mind are the actual mathematical statements that appear in the theories, and not the loosely stated ideas that predate the theories. So the arguments are usually not wrong. They just use annoyingly inappropriate terminology.

Otherwise, yes, the inequality is best understood geometrically, thinking of functions as vectors.

 

[LostConjugate]

I have heard mixed opinions about hidden variables. It is beyond my current knowledge as I am not even taking QFT or QED yet, just core. Some interesting stuff you posted though.

 

[humanino]

I have heard mixed opinions about hidden variables.

Please note that I am not lobbying for any interpretation. I only point out that Heisenberg's inequality does not mathematically imply fundamental uncertainty.

 

[ansgar]

Enlighten us ansgar.

so you don't know how to derive it but still uses it as if you know what it is?

there are library items here on the forum which is really good, have a look at them.

 

[SpectraCat]

are you cute?

I said that HUP has to do with measurements and not energy levels, I had no time to give a full derivation of HUP

Yes, and I said that you were wrong, and pointed out why I think so. Do you have a rebuttal?

 

[ansgar]

Yes, and I said that you were wrong, and pointed out why I think so. Do you have a rebuttal?

I can derive the existence of discrete energy levels without the HUP if that is what you mean by rebuttal?

 

[LostConjugate]

I can derive the existence of discrete energy levels without the HUP if that is what you mean by rebuttal?

I thought we were using HUP to prove how atoms can be stable. The OP question changed after a couple posts.

I was saying that boundary conditions are the reason for discrete energy levels.

 

[sokrates]

I can derive the existence of discrete energy levels without the HUP if that is what you mean by rebuttal?

HUP is an integral (maybe the most fundamental) part of quantum theory, so much so that without it Quantum Mechanics becomes "Classical Mechanics"... And needless to say, you don't have any discrete levels in classical mechanics. So, no, without HUP you cannot "derive" anything..
(Solving a wave equation in 1D with the simplest boundary condition does not qualify for a theory, if that's your secret derivation!..)

 

[ansgar]

I thought we were using HUP to prove how atoms can be stable. The OP question changed after a couple posts.

I was saying that boundary conditions are the reason for discrete energy levels.

Atoms are stable without hup

 

[ansgar]

HUP is an integral (maybe the most fundamental) part of quantum theory, so much so that without it Quantum Mechanics becomes "Classical Mechanics"... And needless to say, you don't have any discrete levels in classical mechanics. So, no, without HUP you cannot "derive" anything..
(Solving a wave equation in 1D with the simplest boundary condition does not qualify for a theory, if that's your secret derivation!..)

yes it as an important thing, but people are miss using it since they don't really know where it comes from, can YOU derive it?

you do have quantization in CM, nodes in a flute, string or whatever!

you don't need and should not use HUP to derive energy levels in an atom and why it is stable! HUP is the standard deviation of measurment of momentum times the standard deviation of position is greater than hbar/2.

all we can say about the hydrogen atom is about standard deviations of it's energy levels etc, NOT absolute values!

Do you want me to call ZapperZ to step in here and judge who of you and I are correct?

 

[sokrates]

yes it as an important thing, but people are miss using it since they don't really know where it comes from, can YOU derive it?

you do have quantization in CM, nodes in a flute, string or whatever!

you don't need and should not use HUP to derive energy levels in an atom and why it is stable! HUP is the standard deviation of measurment of momentum times the standard deviation of position is greater than hbar/2.

all we can say about the hydrogen atom is about standard deviations of it's energy levels etc, NOT absolute values!

Do you want me to call ZapperZ to step in here and judge who of you and I are correct?

Without HUP, your electrons become purely classical, "guitar" frequencies, and your theory falls apart very quickly.. I think this is rather obvious, in fact in many contexts $\hbar \rightarrow 0$ limit is invoked to re-derive classical formulas from QM. [ And that's what you are doing by completely disregarding HUP.]

"Just" stealing Schrodinger equation from QM, and doing a few calculations fails pretty quickly.. It's simply not QM anymore, without HUP..

 

[saaskis]

Of course ansgar is correct that one does not explicitly need HUP to "derive" the stability of atoms. But if one wants to adopt this line of thought, one doesn't actually need HUP for anything, since it's fully included in the quantum theory itself. But HUP is still such an important relation, since it compresses a big part of quantum "weirdness" into a small and simple relation. Therefore, HUP is almost always an excellent starting point when one wants to understand the simplest effects of quantum mechanics heuristically. So in my opinion, if one wants to take a first step into understanding QM by considering HUP, there is nothing wrong with that.

Feynman once said something about how important it is to know many ways to derive one result.

 

[ansgar]

Of course ansgar is correct that one does not explicitly need HUP to "derive" the stability of atoms. But if one wants to adopt this line of thought, one doesn't actually need HUP for anything, since it's fully included in the quantum theory itself. But HUP is still such an important relation, since it compresses a big part of quantum "weirdness" into a small and simple relation. Therefore, HUP is almost always an excellent starting point when one wants to understand the simplest effects of quantum mechanics heuristically. So in my opinion, if one wants to take a first step into understanding QM by considering HUP, there is nothing wrong with that.

Feynman once said something about how important it is to know many ways to derive one result.

yes of course HUP can be used to MOTIVATE certain (all) results in QM, but HUP is not special to QM either - it exists in signal analysis etc. as well! Thus the key concept to all QM "understanding" lies in what postulates - i.e. which interpretation of QM - one adopts.

The reason for why the energy levels are discrete in atoms are due to properties of the wave equation and that one assigns energy to those eigenfunctions. For other waves we don't think that discretezation is funny or weird, like standing waves on a drum for instance. All the phenomenon in QM exists in classical mechanics, but the interpretation is different.

 

[ansgar]

Without HUP, your electrons become purely classical, "guitar" frequencies, and your theory falls apart very quickly.. I think this is rather obvious, in fact in many contexts $\hbar \rightarrow 0$ limit is invoked to re-derive classical formulas from QM. [ And that's what you are doing by completely disregarding HUP.]

"Just" stealing Schrodinger equation from QM, and doing a few calculations fails pretty quickly.. It's simply not QM anymore, without HUP..

HUP exists classically as well.

so now I have one more guy supporting me that HUP is "heuristically" i.e. not really fundamental...

I am a teacher at university in QM classes, who are you?

 

[saaskis]

HUP exists classically as well.

There definitely are analogues of HUP in classical physics as well, but we are talking about point particle mechanics here. And in classical point particle mechanics, there is no HUP. So when one applies HUP to a point particle, one is doing QM, and it is this HUP that everyone is talking about here.

 

[ansgar]

There definitely are analogues of HUP in classical physics as well, but we are talking about point particle mechanics here. And in classical point particle mechanics, there is no HUP. So when one applies HUP to a point particle, one is doing QM, and it is this HUP that everyone is talking about here.

HUP for waves in classical mechanics exists.

QM is sort of a mixture of point particle classical mechanics and waves, hence the old name "wave mechanics"

 

[SpectraCat]

HUP exists classically as well.

so now I have one more guy supporting me that HUP is "heuristically" i.e. not really fundamental...

I am a teacher at university in QM classes, who are you?

No, he only said that the HUP is not unique ... he seems to me to be arguing that it is fundamental to Q.M. ... read that last line of his post again:

It's simply not QM anymore, without HUP".

In my experience it is much easier to get students to understand and anticipate many quantum phenomenon starting from the HUP. Take zero point energy for instance ... early in my class I was able to get my students to predict the existence of zero point energy, just by asking them to think about what must happen in terms of the HUP when a particle is confined in a particular region of space. Similarly, I used the HUP to explain why, for a given system, the energy of an eigenstate increases with the number of nodes in the wavefunction. Another great example is the triple Stern-Gerlach experiment, which is another great illustration of the fundamental nature of the HUP, and how some of it's ramifications were directly evident in very clear ways in the early experiments that established the foundations of QM.

I have found that this kind of qualitative understanding really helps the students to keep up with the course material, and to keep it as part of their general knowledge when they leave the course.

 

[sokrates]

HUP exists classically as well.

so now I have one more guy supporting me that HUP is "heuristically" i.e. not really fundamental...

I am a teacher at university in QM classes, who are you?

I am a graduate student so you think that makes your point stronger?
I think this is a pointless debate. I think it's not my duty to "convince" you that HUP is fundamental to QM and without it, all the interesting quantum phenomena wouldn't exist.

I honestly don't know what you are arguing against here besides, well, this one's almost verbatim from Feynman, so who are you?

 

[espen180]

Ad hominem arguments won't resolve anything.

 

[ansgar]

I am a graduate student so you think that makes your point stronger?
I think this is a pointless debate. I think it's not my duty to "convince" you that HUP is fundamental to QM and without it, all the interesting quantum phenomena wouldn't exist.

I honestly don't know what you are arguing against here besides, well, this one's almost verbatim from Feynman, so who are you?

I would say that HUP is fundamental but not the ONLY and MOST fundamental thing. The MOST fundamental thing is the wave function which is used to DERIVE HUP, i.e. HUP is not more fundamental than the schrödinger wave equation for instance (which is used to derive energy levels in eg. the hydrogen atom). So that was a good argument from my side, HUP is not the most fundamental thing and that interesting phenomena do exists without using it.

HUP gives the standard deviation time standard deviation in position > hbar/2 i.e an upper limit - is that really so hard to accept? Now since it gives the limit of things, it can be used heuristically to give a FEELING and orders of magnitude estimates. It is not strange that HUP gives "ok" results, it only shows that QM is self consistent.

So i stress again, the most fundamental thing is the wave function - without it - no interesting phenomena would exist ;)