The discussion centers on the existence of quantum levels, specifically addressing the role of boundary conditions and the Uncertainty Principle in quantum mechanics. Participants debate the implications of the Hamiltonian energy eigenvalue equation, emphasizing that boundary conditions dictate the lowest energy states. The conversation also critiques the interpretation of the Uncertainty Principle, clarifying its independence from energy levels and its foundational role in quantum mechanics. Ultimately, the consensus is that boundary conditions are crucial for understanding the stability of quantum states.
PREREQUISITESStudents and professionals in physics, particularly those focusing on quantum mechanics, theoretical physicists, and anyone interested in the foundational principles governing atomic stability and quantum behavior.
LostConjugate said: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 said: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?
d-wat said:what I'm asking is what causes the existence of the the thing that keeps electrons in place?
LostConjugate said: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 said:stop miss using the uncertainty principle! it has to do with observations not energy levels
Have you done your QM class boy??...
ansgar said:stop miss using the uncertainty principle! it has to do with observations not energy levels
Have you done your QM class boy??...
LostConjugate said:Huh?
http://www.physics.sfsu.edu/~greensit/book.pdf
Page 90 "Why the hydrogen atom is stable"
SpectraCat said: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.
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.ansgar said:those derivations are just heuristic pseudo arguments found in introductory books
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 conditionWhat 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.
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.
"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 said:Enlighten us ansgar.
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.LostConjugate said:It is because of this inequality and functions having the properties of vectors that we have uncertainties right?
Otherwise, yes, the inequality is best understood geometrically, thinking of functions as vectors.Fredrik said: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.
Please note that I am not lobbying for any interpretation. I only point out that Heisenberg's inequality does not mathematically imply fundamental uncertainty.LostConjugate said:I have heard mixed opinions about hidden variables.
LostConjugate said:Enlighten us ansgar.
ansgar said: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
SpectraCat said:Yes, and I said that you were wrong, and pointed out why I think so. Do you have a rebuttal?
ansgar said:I can derive the existence of discrete energy levels without the HUP if that is what you mean by rebuttal?
ansgar said:I can derive the existence of discrete energy levels without the HUP if that is what you mean by rebuttal?
LostConjugate said: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 said: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 said: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?
saaskis said: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.
sokrates said: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 <br /> \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 Schrödinger equation from QM, and doing a few calculations fails pretty quickly.. It's simply not QM anymore, without HUP..
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 said:HUP exists classically as well.