Calculating the Planck Length

[Dark85]

Homework Statement

I made an attempt to calculate the Planck Length

Relevant Equations

E=hf, E=mc^2, Formula for Schwarzschild radius

I was watching a video by Brian Cox on the Planck length. At 8 minutes and 31 seconds, he begins to talk about how one, in order to make an observation at size the Planck length, must use light of a tiny, tiny wavelength i.e. high energy photons(E=hf). He explains that so much energy goes into observing it that it forms a black hole. So I did some simple calculations on whatever equations I know to try to derive the formula for the Planck Length.

I assumed in order to make an observation at the Planck Length, one must use electromagnetic radiation of some wavelength. Assume my wavelength or$λ=2l$(I will explain my reasoning later in the question).

Now the wavelength of light is related to speed of light and frequency is $λ = c/f$

But
$$
E = hf
$$
$$
f = \frac{E}{h}
$$
$$
λ = \frac{ch}{E}
$$
$$

Now I just took E=mc^2 and I replace λ by 2l.... Hence

$$
$$
2l = \frac{ch}{mc^2}
$$
$$
2l = \frac{h}{mc}
$$
$$
m = \frac{h}{2lc}
$$

Now Brian Cox stated that the energy to required to observe at such small scales will result in the formation of a black hole. So I assumed that the Schwarzschild radius of such a black hole would be equal to l. This is why I assumed the wavelength to be 2l. If l is really the smallest length that is physically possible, then assuming the size of the object is l would be incorrect as it's radius would become l/2, which is lesser than the smallest possible length. Also,

$$
l = \frac{2Gm}{c^2}
$$
$$
m = \frac{lc^2}{2G}
$$
Therefore,
$$
\frac{lc^2}{2G}=\frac{h}{2lc}
$$
$$
l^2 = \frac{Gh}{c^3}
$$
$$
l = \sqrt{\frac{Gh}{c^3}}
$$

But when I looked up the formula for the Planck Length, it showed

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

And as seen, it is the reduced Planck's constant in the formula and not just h.

Obviously, my math relies on very simple equation and it is very likely I might have missed somethings due to my lack of knowledge on the mathematics involved. However, I do have questions:

1. Where does the factors 2π come in to the formula to ensure that it is the reduced Planck's constant and not just h in the formula?
2. Are my assumptions mistaken anywhere (especially in taking $E=mc^2$)?
3. How exactly did Brian Cox come to the conclusion that the amount of energy used will result in the formation of the black hole?

λ=

 

[PeroK]

Note that popular science videos (like the one you linked to) are not a valid source for discussion on here. If you are interesting in learning physics, then I wouldn't take anything Brian Cox says too seriously. Given you are trying to calculate things for yourself (which is commendable), you ought to start learning from genuine sources.

Note that a single photon itself does not have a rest mass, so cannot be or form a black hole. In order to have rest mass from EM radiation, you would have to confine the radiation in some way.

In particular, the idea of a single particle or beam of light being so energetic that it becomes a black hole is completely false. Energy is frame dependent. There is no intrinsic kinetic energy associated with a particular particle or photon. It's all frame dependent. Whereas, a black hole has an invariant (frame independent) description.

 

[PeroK]

In answer to your question, the Planck length, as far as I know, is defined to be:
$$l_p = \sqrt{\frac{\hbar G}{c^3}}$$I think it's something of a coincidence that this is in some sense "the smallest meaningful length". For example, the Planck mass is about $2 \times 10^{-8} kg$. I'm not sure that has any significance.

In any case, a factor of $2\pi$ is not particularly significant in the context of the overall scale of something. If we defined:
$$l_p = \sqrt{\frac{h G}{c^3}}$$I don't see that would make much difference to anything.

 

[anuttarasammyak]

Your estimation would show that we have minimum length, which is order Planck length, for our investigation of space. When we try to see the inside inputting high energy required to do it, the area becomes BH which prohibits information of that area coming to us.