Problem — Science fiction, reducing h and atomic volume

Problem

A science-fiction writer proposes shrinking objects by reducing Planck’s constant $h$. Knowing that the Bohr radius $r_1$ depends on $h$ as $r_1 \propto h^2$, by how much would $h$ need to be reduced (as a percentage) to halve the atom’s volume? Comment on why the proposal remains science fiction.

Solution

Step 1 — From radius to volume. Atomic volume scales as the cube of the radius, and the radius as $h^2$:

$V \propto r_1^3 \propto (h^2)^3 = h^6$

Step 2 — Halving condition. Call $h'$ the new constant. Imposing $V'/V = 1/2$ means

$\left(\frac{h'}{h}\right)^{6} = \frac{1}{2} \quad\Longrightarrow\quad \frac{h'}{h} = \left(\frac{1}{2}\right)^{1/6} = 2^{-1/6}$

Step 3 — Numerical value.

$\frac{h'}{h} = 2^{-1/6} \approx 0{,}891$

so $h$ must be reduced by about

$\ev{1 - 0{,}891 = 0{,}109 \approx 11\%}$

Comment. It remains science fiction because $h$ is a universal constant: it is not a parameter that can be “lowered” locally on one object while leaving the rest of the universe unchanged. Changing $h$ would simultaneously rescale all atomic dimensions, binding energies, wavelengths and even the stability of matter and chemistry — the “shrunk” object would no longer be made of the same matter.

Collegamenti

Argomenti: Quantum physics Concetti: Bohr model Competenze: Limiting-case and science-fiction analysis · Symbolic setup