Lady UncertaintyQuantum Mechanics / Heisenberg’s Limit

Lady Uncertainty is the reason you do not collapse.

The relation on this page is the familiar one: across many identically prepared particles, the spread in position times the spread in momentum can never be less than ħ/2. Heisenberg introduced the idea in 1927 with a thought experiment about a microscope kicking an electron with light; Earle Kennard proved this exact inequality the same year. It is usually told as a limit on knowledge. It is also a source of energy.

Squeeze a particle into a small space and its momentum must spread. Spread momentum means kinetic energy, and a lot of it if the space is small. So confinement costs energy, whether anyone is watching or not.

Now take hydrogen. Classically, the electron should fall into the proton: the closer it gets, the lower its electrical energy, with no bottom. The second line on this page is the quantum estimate. Confining the electron within a distance r costs kinetic energy of about ħ² over 2mr², which grows as r shrinks, faster than the electrical attraction falls. Add the two and the total has a minimum. Find it and you get r equal to 0.0529 nanometres, exactly the Bohr radius, and an energy of minus 13.6 electronvolts, exactly hydrogen's binding energy. The exact agreement is partly luck of the rough estimate, but the logic is sound: atoms have a size because squeezing them further would cost more than it gains.

The same zero-point energy is why helium stays liquid at absolute zero under ordinary pressure. Its atoms are light and attract each other weakly, and the jiggle forced on them by confinement is enough to stop them locking into a crystal.

Lady Uncertainty keeps everything slightly apart. That is what holds matter up.