How was the Planck length (or time) calculated?

[Chen]

Can anyone please shed some light on how the Planck length or Planck time were found? I understand why they have to exist, but what predicts their values? Is it even a prediction or can we actually calculate the Plank length with experiment?

Thanks,

 

[NanoTech]

If the measuring device used to make the measurement
of the particle's position is smaller in dimension than the Planck
scale (i.e. Planck length), would the Uncertainty relation still apply.

Sure.

What effect would such a device have on the outcome of the
experiment?

Now, that's a harder question! Consider first that the Planck
length is about
0.000000000000000000000000000000004 cm.
(I think scientific notation packs less of a punch than regular
notation sometimes.) So it's going to be a problem *making* a
device which does this -- but of course we can just do a
gedanken experiment for free instead! So what happens, in
theory?

Suppose we're interested in localizing an electron (or other
massive particle of spin 1/2). Then the relativistic wave
equation we might try is the (single-particle) Dirac equation.
It's not too difficult to squeeze an electron down to the
size of its Compton wavelength, which is about
0.00000000004 cm.
However, if we squish it down past that then negative-energy
components start appearing! Even in the one-particle theory,
localizing the electron beyond its Compton wavelength
requires some *positron* excitation to do the trick.

This is a hint that the single-particle theory isn't right
either. We need to use quantum field theory to describe such
a tightly enclosed electron! In the QED description, extra
positrons and electrons will start appearing whenever we try
to localize beyond the Compton wavelength. (This doesn't
violate any conservation laws, because whatever is doing the
localizing is interacting with the electron somehow, so the
apparatus picks up recoil and things like that.)

 

[Chen]

I'm afraid you went right over my head. What does the principle of uncertainty have to do with Planck length?

What I'm asking really is: What predicts the existence of Planck length?, Does anything predict its value?, and Is it possible to measure the Planck length experimentally, thereby validating the prediction (if it exists) of its value?

 

[jcsd]

The Planck length is:

$$\sqrt{\frac{G\hbar}{c^3}}$$

The Planck time is:


$$\sqrt{\frac{G\hbar}{c^5}}$$

 

[Chen]

And that's because...?

 

[curious george]

Originally posted by Chen
And that's because...?

First, consider the smallest energy x length that you can have, which is dictated by our knowledge of quantum mechancs:

$$\hbar c$$

Now set that equal to the gravitational energy to get it in terms of the gravitational constant:

equation 1 $$GM^2=\hbar c$$

solve for M and you get the Planck mass:

equation 2 $$M = \sqrt{\frac{c}{G\hbar}}$$

The Planck length can then be found with a little dimensional analysis:

$$GM^2$$ has units of length x energy (M is Planck mass, G is gravitational constant)

$$Mc^2$$ is units of energy (M is again the Planck mass)

and from the equations 1 and 2 you will see that:

$$GM^2 = \hbar c$$

and

$$Mc^2 = c^2 \sqrt{\frac{c}{G\hbar}}$$

And from that you can see that:

$$Planck Length = \frac{GM^2}{Mc^2}=\frac{\hbar c}{ c^2 \sqrt{\frac{c}{G\hbar}}}= \sqrt{\frac{G\hbar}{c^3}}$$

You can do a similar calculation to find the Planck time.

(EDITED to fix a few mistakes)

 

[curious george]

An interesting side note is that, in natural units, where:

$$\hbar = c = 1$$

The Planck mass is just the reciporical of the Planck length.

$$L _{planck} = \frac{1}{M _{planck}}$$