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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!