- #26

JesseM

Science Advisor

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But when you say "dimensional analysis", do you mean there aren't even any rough physical arguments, it's purely a matter of shuffling various constants to reach some physical conclusion?It sounds like your questions are on the concepts of dimensional analysis itself. Dimensional analysis is quite powerful when used appropriately, and just like statistical mechanics, can cut through all the "un-needed details" to give you very useful broad results.

These techniques are used for all kinds of things. While engineers may not always learn where they come from, they use dimensional analysis in many classes. Looking up pressure curves for non-ideal gases, often use scaled pressures and temperatures, etc. and the gases, despite being non-ideal, are now all described almost identically on a graph. Consider Reynolds number, for fluid dynamics etc.

OK, this seems like a more physical argument (just to be clear, when you talk about the energy needed in QM I assume you mean something like the minimum depth a potential well of width L would need in order for there to be at least one bound state for a photon, i.e. if L is the wavelength then a bound state which fits about one wavelength in the well would have energy approximately equal to E=hf=hc/L according to Planck's equation, so the potential must be at least that deep for there to be a bound state). Perhaps a physicist would make a purely dimensional argument when talking to an audience of physicists who would be assumed toJustinLevy said:To help "bridge the gap", I'll attempt to make the dimensional analysis "more physical" sounding to you here.

1] Length scale in classical GR

for black hole of energy E, on order of

[tex] L_{gr} = \frac{GE}{c^4}[/tex]

2] Length scale from quantum mechanics

Due to quantum wavelength

[tex]\Delta x \Delta p \ge \frac{\hbar}{2}[/tex]

to contain in a box of size L requires an energy on order of

[tex]E = \frac{\hbar c}{L_{qm}}[/tex]

these two become comparable at:

Length scale = planck length = [tex]\sqrt{\hbar G/c^3}[/tex]

Reworded to sound more physical:

If we have a huge blackhole, quantum mechanics doesn't have much to say... GR should hold fine. However, for very small length scales, GR could claim there is a blackhole, but quantum mechanics says the very wave nature of particles prevents it from being confined in such a small volume. This is the scale at which quantum effects (wave nature of particles) make the assumptions of GR (classical fields and particles) become incorrect enough that we wouldn't be approximating the real result with GR anymore. We'd need a quantum theory of gravity.

*know*a way to translate this into a more physical argument, but what I question is whether it's meaningful to use a purely dimensional argument if you don't have a more physical argument in mind. How would you tell "good" dimensional arguments from "bad" ones (like a hypothetical argument saying that quantum gravity effects should apply to any particle with a mass smaller than the Planck mass, or that quantum electrodynamics effects only become important for charges smaller than the Planck charge) if you don't have recourse to more specific physical arguments?

Well, if you divide planck's constant by (c*some mass) you get something with units of length, but in purely dimensional terms how do you decide whether to use the mass of an electron or the mass of a proton, since the two differ by three orders of magnitude? Again it seems like you need some sort of physical argument, even if it's one a physicist would find obvious enough to just leave it implicit...JustinLevy said:Test your knowledge:

Classical electrodynamics says there cannot be any stable arrangement of charges. Quantum mechanics disagrees with the assumptions of classical electrodynamics, and luckily for us allows atoms to form. Without solving Schrodinger's equations, using dimensional analysis, can you find what length scale classical electrodynamics breaks down for an electron interacting with a proton, and therefore what size you would expect atoms to be?