Why matter does not collapse

Problem

The ordinary matter we see is made of atoms formed from nuclei and electrons. Why don’t they collapse? Why doesn’t the electron fall into the nucleus, as it should according to classical physics? Argue in terms of uncertainty and discrete energy levels.

Solution

The classical paradox. Classically, the electron, attracted by the nucleus, should radiate energy and spiral inward: the atom would collapse in $\sim 10^{-11}\;\text{s}$. This does not happen: atoms are stable and have a well-defined size ($\sim 10^{-10}\;\text{m}$).

The uncertainty argument. Confining the electron within a region $\Delta x$ requires, by Heisenberg, $\Delta p \gtrsim \hbar/(2\Delta x)$, and hence a confinement kinetic energy

$K \sim \frac{(\Delta p)^2}{2 m_e} \sim \frac{\hbar^2}{2 m_e\,\Delta x^2}$

which grows as $\Delta x$ shrinks. The total energy is the sum of this term (positive, increasing as we get closer) and the attractive potential energy $\sim -k e^2/\Delta x$ (negative). This trade-off has a minimum at $\Delta x \sim a_0 \approx 5\cdot 10^{-11}\;\text{m}$: “falling into the nucleus” ($\Delta x \to 0$) would cost infinite kinetic energy, so it is energetically forbidden.

Discrete levels. Quantisation fixes a ground state ($n=1$) of minimum energy: below it there are no allowed states. The electron cannot lose energy continuously — it can only jump between discrete levels by emitting photons, until it reaches the ground state, where it remains stable.

$\ev{\text{The confinement kinetic energy (uncertainty) + discrete levels prevent collapse}}$

It is the same reason, on a larger scale (with the Pauli principle added), why ordinary matter is “rigid” and occupies volume instead of imploding.

Links

Topics: Quantum physics Concepts: Uncertainty principle · Bohr model Objects: Hydrogen atom