Exploring the Physical Meaning of Planck Units

[jbar18]

I've been looking at the various Planck units, and I'm wondering how they are constructed from the constants involved. Like is there any physical reasoning behind those equations? I've looked all over the place for derivations of the equations or what the actual physical meaning of the units are, but it seems to be that Planck just found that you could combine the constants to give the correct dimensions for the units. Is that all there is to it? Or is there an actual reason for assuming that a fundamental unit of length will involve G, c and h? If someone could explain I would be very grateful.

 

[MathematicalPhysicist]

It's called dimensinal analysis, you don't know for sure if there isn't a mathematical constant which needs to be factored as well, so you look at fundamental physical units which are inherent in the problem.

In case of Planck length we know that the maximum barrier over the speeds of particles is c, so we know it should be included, we also know that at the tiny scales there's a parameter of h cause at the macro h=0, and we also look for gravity effects in this scale, which means G should be there.

Obviously this is a rough estimate of the smallest length, if there something like that.

 

[jbar18]

Thanks for the response MathematicalPhysicist. I was just wondering whether there were, in particular, any thought experiments or anything which point to the equations, but it seems that they are just put together to make the dimensions work as an estimate. I guess it's a way of finding "nature's units" by combining real constants, I'm just having a hard time thinking about what the equations actually mean (if anything).

 

[Bill_K]

It's more than just dimensional analysis, there's actually a reason behind it. It's the scale at which the Compton wavelength is equal (ignoring a factor of two) to the Schwarzschild radius.

GM/c² = ħ/Mc

Solving for M you get M = √ħc/G, the Planck mass. The Planck length is then

GM/c² = √ħG/c³

Significance:

The Compton wavelength is the distance scale at which the zero-point energy of a confined quantum particle is equal to its rest mass.

The Schwarzschild radius is the distance at which the gravitational potential energy of a particle is equal to its rest mass.

All three (zero-point energy, gravitational potential and rest mass) become equal at the Planck scale.

 

[bobie]

It's the scale at which the Compton wavelength is equal (ignoring a factor of two) to the Schwarzschild radius.
[2]GM/c² = ħ/Mc
Solving for M you get M = √ħc/G, the Planck mass.
The Planck length is then
GM/c², L= √ħG/c³

Hi, could anyone please clarify these points:
- why are we allowed to ignore the factor of two?
- isn't the formula h/mc?
- why should the unit of length L be equal to G/c² * M?
- is that the real genesis of the units? In 1900 the Schwartzschild radius was not known.

- one last problem : the r_s of the electron is much smaller (10^-50m) than its λ_C (10^-12m), and even smaller than L_p (10^-35m)