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The Planck Length

An exception is the overly confined particle. Even a massless photon, if confined to a small enough region, will have such an enormous energy of confinement [from Eq. (17)] that its effective mass

$$m_{\rm eff} \; = \; {n h \over 2 \ell c} .$$ (24.20)

(from $E = m_{\rm eff} c^2$) will be big enough to make $\ell$ smaller than the Schwarzschild radius! Using $m_{\rm eff}$ in the formula (19) for R_S and setting $\ell = R_S$ gives an approximate formula for the PLANCK LENGTH

$$\ell_P \; = \; \sqrt{ h G \over c^3 } .$$ (24.21)

If you try to confine any particle (even a photon) to a region smaller than $\ell_P$, it will cause a gravitational collapse into a black hole. I.e. you can't do it. This is where quantum mechanics is certain to break down. Want to do some leading edge Physics theory? Quantum gravity is a good place to start.

By the way, the above handwaving derivation simply explains why you can't confine a particle to a prison of dimensions smaller than the Planck length. It says nothing about restrictions on an empty prison, nor does it make any claims about any ``grainyness'' of spacetime. If there is such quantization of space and time, you will have to construct a different argument for its existence. People do. But hey, this is weird enough!