timo

Pardon the sarcasm, but: What's that rule good for?

"far away" is quite a relative statement. The distance is roughly 0.1 nanometers, i.e. 0.2% of the wavelength of visible light. I believe in many cases that would count as "the electron and the nucleus are practically at the same position". That is also the cases you might expect that often it is the center-of-mass that interests you. You are of course right that this is not always the case. Sometimes, the relative location of proton and electron is important. So please do not misunderstand my comments as saying that you always treat the electron-proton bound state as a single point-sized Hydrogen. That is not the case. What I tried to argue is that in the context of asking for de Broglie waves, plane matter waves, the concept can only be applied to the center-of-mass part of the total wave function. The relative positions of electron and proton are still relevant in other scenarios. For example, the spectral line structure of hydrogen (the light frequencies absorbed and emitted by it) have their origin in the relative-position part of the hydrogen. The center-of-mass part merely adds some blue- or redshift to it. Also, the common separation into a center-of-mass and a relative position has no physical significance. It is merely chosen for practical reasons (to make the math clearer and to be able to attribute phenomena to different parts of the wave function). Purely physically, you could treat the system as an electron and a proton with an interaction between them. You just would not be able to make much sense out of the mathematical notation of the wave function you get.

Exactly. That is why I recommend accepting these "take home" messages, even if they are incomplete: - The hydrogen atom is usually described in terms of the center-of-mass of electron and proton and their relative position (and not in terms of the two particles' positions individually, which would technically be possible). - In this picture, the center of mass behaves as a single particle with a mass equal to the sum of electron and proton mass. - This implies a corresponding de Broglie wave for the center of mass. - The relative position is associated with a more complicated wave function, which is not described by a de Broglie wave. - The relative position part adds to the total energy. But it does not influence the center-of-mass part. - It is not automatically true that the whole hydrogen atom is well-described by its center-of-mass alone. But if you ask for a de Brogile wave (-length) of a hydrogen atom, the de Broglie wave (-length) of the center-of-mass part is what most people would mean by this (because the relative part would usually not be called a de Broglie wave).

The total wavefunction of the system has two parts: The center-of-mass part and the relative-distance part. Both contribute to the energy of the system. The former behaves as a free particle (-> de Broglie wave), the other is what is commonly meant by "energy levels of the hydrogen atom". The two parts add up to yield the total energy of the hydrogen atom. Also note that the scenario is a bit more complicated than you seem to imagine it, but the proper way to understand that would require a bit of math. The following page may be a bit too advanced, but it describes what I mean by the "two parts" and demonstrates how their energies "add up": http://farside.ph.utexas.edu/teaching/qmech/lectures/node58.html#stwo . The relevant equation is (424), which explicitly shows that energy is a sum of center-of-mass motion, relative motion, and potential energy due to relative position.

There is no physically "real" correct answer to what the de Broglie wave (-length) of a composition of particles is. But there is an mainstream accepted community-wide agreement on how to treat that particular case: When you start developing the quantum mechanical wave-function of the hydrogen atom, the very first step is a change in coordinated: The coordinates of the proton's and electron's center of mass (which effectively is just the location of the proton) and the deviation of the two particles from their common center of mass. The solution for the full system then separates into a wave function of the center of mass and the relative positions of proton and electron. The former is what you could call "the full atom". That's the thing you are after, and its de Broglie wave should behave as if the particle had a mass equal to the sum of the individual masses (although I am not 100% sure of that at the moment). The other part, the one you are not after, is what is treated in QM texts about the hydrogen atom. In short: You treat the hydrogen atom as a single object in its own right.

You can PDFify it (and combine with other articles) using the Wikipedia book creator: https://en.wikipedia.org/w/index.php?title=Special:Book&bookcmd=book_creator

Iirc, the Banach-Tarski paradox is an effect of the axion of choice which is required/introduced in the context of real numbers. Rational numbers would not fall under this paradox, I think. Rational numbers are not quantized.

Not a native speaker. But maybe a "survey of N" does not necessarily imply that N of the survey group gave a valid feedback. In field tests with customers you can usually be happy if half of the survey group fill out a final evaluation form. In medical trials, there is even explicit statistical methods telling you how to deal with participants that drop out (or drop dead) during the experiment.

Not really. I wasn't doubting that particles in a solid have less tendency to move apart from each other. I was doubting that their speed (in the sense of magnitude of velocity) is lower. In fact, increasing the attraction seems quite compatible with the particles still having similar speeds but not separating anymore.

Why would particles that are packed denser have lower speeds? They sure have smaller mean free paths. But why wouldn't they bump around just as fast as if they had more space?

Saying that a cup of coffee does not fall down because the table "pushes it up" is quite a common statement, I think. I'm pretty sure at least one of my teachers at school explained it like that (in the context of "the reason why gravitational force does not cause this to fall is because there is another force canceling it). The other half of the statement ("nothing holds us down") is debatable. Sounds more like semantics to me, and I doubt any insight is gained from supporting, discussing or denying the statement (although my school teachers might have had a different view on this ). But calling it "crackish" seems a bit inappropriate to me, especially when considering that it was in reply to a really strange comment.

I'm not quite convinced that reading up on GR really helps understanding your statement.

Incidently, I spent half of the day looking through data of how well my simplified model (of a non-physics system) describes reality. At some point I realized that my excuses why data don't match my model became a bit ridiculous, so I ended up throwing them all into the trash bin (the excuses, not the data) and going with "doesn't work, but we don't have a better alternative so we stick to the model" . Oh, the joys of contract-based research where every project has to be a great success by definition.

The effect of gravity is attributed to curvature in space-time. The graviton is a wave that alters space-time (not quite sure if it only bends and stretches or actually curves). The smallest possible of such waves, in a sense. Often, this means a wave that alters an otherwise-flat space-time. There is, to my knowledge, no known case of gravitational interaction between sub-atomic particles (*). On the known scale, where gravity is observed regularly, gravitons do not play a role. In fact, not even "large" gravitational waves have been detected directly, up to date (**). On the sub-atomic level, we do not really know how/if gravity works. But you will readily get a lot of ideas about it if you ask around. Many of them will involve statements claiming gravitons are the ultimate source that accounts for all curvature. The reason is that this is a direct extrapolation from a simplified perturbative treatment of e.g. electromagnetism. And extrapolating from what has worked somewhat well in other cases is not exactly the worst kind of guess one can make. (*) Please note: I do not mean to question the mainstream working hypothesis that there is a tiny interaction. I am just saying that to my knowledge no experimental observation for this exists. (**) Haven't had the time to read the latest hype about gravitational waves having been detected. But my blind guess is that they detected patterns that can only be described if there are gravitational waves. Not that their sensors actually reacted to a gravitational wave.

That's the crucial point. To put it differently and referring to the thread title: There is no real time order between the two events (and thus, strictly speaking, no "reverse order"). In relativity there are two ways how events can be related. They can be causally related, which means that one event is always happening after the other - for every observer. This allows to claim they "really" have an order. Or they may be non-causally related, meaning not only that their order may depend on who observes them but also that they can not influence each other.(*) A typical construction in this context is the light cone (https://en.wikipedia.org/wiki/Light_cone). Events that lie within each others' light cones (future or past light cone) are causally related. Events that do not, a non-causally connected. The case that event A lies in the light cone of event B but event B does not lie in the light cone of event A can not occur. (*) Note: Strictly speaking, on top of the time-like and space-like relations I mentioned (I just called them causally and non-causally because that names are much more appropriate) one usually distinguished light-like relations that mean a relation where only information traveling with the speed of light can connect the two events. In the context of this post, that is just a special case of causally connected events.

The infinite amount of numbers in the interval [0;1] does not automatically imply that the number 2 lies within this interval. It doesn't even make it probable that a randomly-chosen value lies within this interval.

That questions sounds far more suited to be asked to the moderator via private messaging than to be discussed in a thread with people who cannot even see the post is question.

Well then, slow down: What is your actual problem? Finding the "x" that solves the equation?

Yes.

My guess is that with no matter at all no one would care. Locally matter-less regions are called vacuum, and generally considered as being part of space, implying that space exists in these regions(*). (*) But of course the very term "region" implies space, so the statement is circular reasoning to some extend.

No. EDIT: Too late. My elaborated statement was in reply to the very detailed original question, of course. Not to Strange's post.

I'm not sure that "the fundamental concept is [a cryptic arrangement of greek and latin letters without any trace of explanation]" really helps much. The equations being syntactically wrong doesn't really help, either: The first equation shold probably have all-lower indices and the 2nd one has a misplaced sum-sign (assuming the unlabeled sum is supposed to sum over a and b). Minor typos, though. Not explaining what the symbols stand for is the big problem. On topic: I am not sure that in physics there is that one statement from which everything else derives. Even though that's admittedly how physics is often sold to students. One might be tempted to say that some hocus-pocus statements like "spacetime is described as a manifold with intrinsic geometrical properties" or "the speed of light is the same for all observers" were such fundamentals. But that's just ignoring that those statements need of a lot of other assumptions, a framework to "live in", to even have any meaning.

I think you should take into account that faizan is, if I understand correctly, an engineer (electrical or renewables, I guess). I'm not sure that "Experimental or Theoretical Condensed Matter Physics" is the right direction to point to. But then, I'm not quite sure how thermodynamics for engineers can be anything else than applied research. I do equate the term with "Statistical Physics", which already gives a hint of where the field is located academically. @faizan: To get an impression of what we are actually talking about: What are those "most institutes" doing applied research that you talk about?

Didn't check the math (again ), but when plugging into equation b you probably want to plug in z for z, not for y. If you want to become a physicist (and perhaps also if you want to become an engineer) then learning tex is certainly recommended: pretty much every physicist knows and uses tex, so you can as well pick up that skill early. I'd say that solving systems of linear equations is frequently done in physics, even though I cannot pinpoint a sensible example at the moment. There's two straightforward generalizations of it that are an asset in every physicist's math toolkit (which is why I tend not to realize the special case of systems of linear equations). The first one is linear algebra, which deals with structures called "matrices" that are similar to systems of linear equations. The other one is quite generic: Often, you have a system involving several unknown variables you want to know. What you often do is that you look for a sufficient number of known relations between the variables to be able to solve for their values. That is very similar to the concept of sets of linear equations, but generally the relations do not have to be linear.

There is at least one mistake in the calculations you made (I stopped reading at some point). But the approach is correct.