joigus

Wave functions of stationary states work pretty much as the density/current of a stationary fluid. Consider the flow lines of a fluid in a stationary flow state. At every point in the fluid, the velocity field is well defined, has a direction and a speed, even though nothing seems to be moving on the whole. More mathematically: If your stationary state is represented by wave function \( \psi_{n,l,m,s}\left(\boldsymbol{x},t\right)=e^{-iE_{n,l,m,s}t/\hbar}\Psi_{n,l,m,s}\left(\boldsymbol{x}\right) \), your 'cloud' of probability would be independent of the time-varying phase factor: \[ \varrho\left(\boldsymbol{x}\right)=\left|\Psi_{n,l,m,s}\left(\boldsymbol{x}\right)\right|^{2} \] The whole situation would be static, and yet, it would have an associated velocity field, which mathematically is given by the Fourier transform of the amplitude, \[ \hat{\psi}_{n,l,m,s}\left(\boldsymbol{p},t\right)=\frac{1}{\left(2\pi\hbar\right)^{3/2}}\int d^{3}xe^{i\boldsymbol{p}\cdot\boldsymbol{x}/\hbar}e^{-iE_{n,l,m,s}t/\hbar}\Psi_{n,l,m,s}\left(\boldsymbol{x}\right)=e^{-iE_{n,l,m,s}t/\hbar}\hat{\Psi}_{n,l,m,s}\left(\boldsymbol{p}\right) \] So the distribution in momenta doesn't depend on time either: \[ \varrho\left(\boldsymbol{p}\right)=\left|\hat{\Psi}_{n,l,m,s}\left(\boldsymbol{p}\right)\right|^{2} \] This is only valid for stationary states. I hope that answers your question. It's good to see you around.

What is gravity? As @MigL and @beecee said, ultimately we don't know. You can model it as force at a distance, à la Newton, but that doesn't work for rapidly changing or relatively strong (stellar) fields. There comes general relativity to the rescue. You can model it as geometry of space-time, but that leads to a couple of problems: 1) Horizons are entropic (they hide information) 2) Gravity is non-renormalisable at arbitrarily strong fields (ultraviolet limit, high-energy collisions), because it's dimensionally bad-behaved. Problem 1) is both conceptual: What is this geometry with an entropy/temperature; what are the hidden degrees of freedom? And it is also wanting in mathematical/logical consistency: Since Hawking we know that black holes must evaporate if gravity is a quantum field, so microscopic information disappears ==> Distinctions between trajectories disappear ==> Predictability vanishes at too fundamental a level. Black holes are so interesting because they are quantum objects and they are general-relativistic objects; so many people hope they will show us eventually what's wrong with the present picture. A very interesting change in the mind frame of physicists took place in the 20th Century: Renormalisation. What is renormalisation? In very general terms, it's the realisation that physical problems look one way or another depending on the scale and the range of phenomena at which you wish to describe things. Consider a wooden stick. What is it? Newtonian mechanics considers it as a rigid body, which is described by 6 real variables --typically the position of its centre of mass and three orientation angles. But what if you want to consider situations like shooting at it, deforming it, breaking it into pieces? (higher energies, changes at small spatial scale). Then your parametrisation is no longer useful, and you need to take into account interatomic interactions. The motto for this loss of innocence is: My parametrisation of the physical system is not the physical system; it's just my parametrisation of the physical system.

I think I speak for most everyone here when I say that intellectual honesty is high up there; above erudition or any similar razzle-dazzle. +1.

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That's LaTeX. https://www.scienceforums.net/forum/99-the-sandbox/

Great performances, but time has not been kind to poor old Moby Dick.

Yeah. Everything OK IMO.

Number-one step should be dating the rock, I think. For dating rocks in the order of billions y.o. a trusted method is based on ratios U/Pb in zircon crystals. If the rock is a meteoroid, it should be as old as the solar system. https://www.amnh.org/learn-teach/curriculum-collections/earth-inside-and-out/zircon-chronology-dating-the-oldest-material-on-earth

(my emphasis.) Huh? The atmosphere sounds like a fun place --I don't mean the bacteria.

Let's give them a call: Hi, @Enthalpy.

This hypothesis was very much talked about as far as decades ago. If I remember correctly, it was on a Scientific American issue.

Great answer IMO. Very interesting aspects have surfaced here that no doubt played a role in evolutionary history of humans and their brains. But as Ken points out --in my own words-- we must keep in mind that evolution is multi-factor, intricate, arguably chaotic. A particular development that could be advantageous if you, as an animal, have a lifestyle 'on-the-run', could be detrimental if you are sedentary; advantageous if you're a predator, detrimental if you're more of a gatherer, ineffectual if you're a scavenger; advantageous if you live in the tropics... Etc. You get the picture. I think what the OP suggests as a possibility could well have happened --had one or many of the circumstances that constrained human way of life been different. Maybe modern humans would be like pandas, which have some of the most extreme ratios in size between infants and adults.

That's not general. That's only true in non-relativistic classical mechanics and in gauge field theory. Not in GR, not in relativistic particle mechanics.

That's why your depiction of a sombrero potential superimposed to a gravitational singularity doesn't make any sense: One makes sense with the spatial radius, \( r \) as a variable; the other with the 'radius' of the field variable \( \left| \varphi \right| \). Unless I've misunderstood you really badly, but as I see no equations...

For balance of what? In the standard model, the Higgs field is a constant, which corresponds to a possible vacuum. The famous sombrero curve is not the field; it's the Higgs potential, which depends of the field. The X-axis is the field variable, and the Y-axis is the potential energy. Space doesn't even appear in the picture unless you excite the field locally. And, as Swansont said, what singularity? The sombrero potential has no singularity. The gravitational field of the Earth has no singularity either. And so on, and so on...

What mass are you talking about? Inertial mass or gravitational mass? They are very different concepts.

If they were, we might as well stop saying 'no' altogether. I don't think you want to do that.

Agreed. I suppose what you wanna do here, @NTuft, is something like considering a complex variable, \( z \), differentiating by it, \[ \frac{d}{dz}\left(\frac{1}{2}z^{-1/2}\right)=-\frac{1}{4}z^{-3/2} \] and then substituting, \[ \left.\frac{d}{dz}\left(\frac{1}{2}z^{-1/2}\right)\right|_{z=i}=\left.-\frac{1}{4}z^{-3/2}\right|_{z=i}=-\frac{1}{4}i^{-3/2} \] But be careful; you may be re-discovering complex calculus. The connection to number theory is something I'm missing. I'm very skeptic to there being a connection between physics/chemistry and number theory. There is a connection between physics and complex calculus, provided by harmonic functions in two-dimensional problems.

Exactly my feelings. Unfortunately there's an army of people like this in high places.

This looks like no hype at all. My feeling is that this has the makings of a real breakthrough in quantum computing. The possible implications for investigations in climate models, protein-folding, virology models, etc. are mouth-watering. I can't wait for the moment when this chip is finally built. https://phys.org/news/2021-08-critical-advance-quantum.html?fbclid=IwAR2mj_PP9NqXylVALnJQSbysOxzwF4HnU9_zFeNVTBiflnSrTq8phZdNUe0

I agree. This is not to say that physics and mathematics don't have interesting connections. They do, and they surface from time to time. But in the way the OP is dealing with it, I find it almost impossible to fathom what the proposed connection actually is. Again, I'm missing a clear statement of what is supposed to be the conjecture. It's more like a purée of mathematical, physical, and chemical terms.

This argument is incorrect because I was in a hurry. I will try to add the proper disclaimer later. Electrons cannot be probed with collision energies corresponding to wavelengths smaller than their Compton wavelength --I think that's the correct argument.

(My emphasis.) This video is famous and has been going around these forums for quite a while. I think it's related to your problem: Light has no fundamental length parameter characterizing it --unlike electrons, or protons, for example--. Why? There you are. Electrons cannot come in wavelengths much smaller than their so-called Compton wavelength. If you try to do so, you don't get smaller-wavelength electrons; you get more electrons and positrons of larger wavelengths than the electron's Compton wavelength. But photons behave differently. Adding to what Swansont has said, wavelength is a frame-dependent quantity. If you had a photon with a certain wavelength in one frame, it would have a different wavelength in another frame. But then you can ask again, why?

Who would consider such a stupid thing? I mean you do, are doing, \[ \frac{d}{di}\left(\frac{1}{2}i^{-1/2}\right)=-\frac{1}{4}i^{-3/2} \] So you seem to be considering the imaginary unit, \( i \), as a constant. variable. That shows that you don't understand what differentiation is about. No target in sight.

I wanted to fire away real bad, but I didn't spot any physics. No chemistry either. And the only maths is wrong. You can't consider increments of a constant.