Concept

Bohr sommerfeld quantisation — where it appears

The old quantum rule that a periodic motion is allowed only if its action round one cycle is a whole number of Planck constants. Corrected by a half for each pair of turning points, it gives the energy levels of many systems to good accuracy from classical orbits.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The area each quantum state is allowed. Phase-space orbits of the four lowest quantum levels, computed as eigenvalues of the discretised Schrödinger equation, for a harmonic well (left) and a box with hard walls (right), in units ħ = m = 1. Each orbit encloses an area that is a whole number of Planck constants plus a fixed offset: for the harmonic well 0.500, 1.500, 2.500, 3.500 h — n + ½ — and for the box 1.000, 2.000, 3.000, 4.000 h — n + 1. The harmonic orbit turns round smoothly at both ends and loses a quarter of a cycle at each; the box orbit is reflected sharply by each wall and loses half a cycle there.

The quarter cycle a turning point costs

The old quantum rule said an orbit's action is a whole number of Planck constants. It gets the harmonic oscillator wrong by exactly half a unit, the box by exactly one, and a neutron bouncing on a mirror by three quarters. Each is the same correction: a wave loses a quarter of a cycle every time it turns round smoothly, and half a cycle every time it hits a wall.

mechanics · Least action $Three orbits that are one orbit, rescaled. Three orbits round the same mass, each an ellipse of eccentricity 0.6, with sizes in the ratio 1 : 2 : 3, integrated from the inverse-square force. The dots mark twelfths of each orbit's period, at the same fractions of the way round. The shapes are exact enlargements of one another and so are the motions, with time stretched as size to the power 3/2: the measured periods are 1.000, 2.828, 5.196 times the smallest, against 1.000, 2.828, 5.196 from the rule. That is Kepler's third law, obtained without solving for any orbit.$

The third law that falls out of a change of scale

Kepler spent a decade of tables to find that the squares of the planets' periods go as the cubes of their distances. The law can be had in three lines, without solving for a single orbit. Gravity's potential energy halves when distances double; stretch every length of an orbit by some factor and every time by that factor to the power three-halves, and the action changes only by a constant, so the stretched orbit is an orbit too. The same move gives the swing of a spring that keeps time at any amplitude, the virial theorem that makes a star heat up as it loses energy, and the spacing of quantum levels.

mechanics · Least action

Named alongside it

The objects these essays reach for when they reach for this one.

Action Harmonic oscillator Kepler third law Lagrangian Maslov index Phase space Principle of least action Quantum bouncer Scaling Semiclassical approximation Symmetry Turning point

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