Reduced Planck constant

The reduced Planck constant, written ℏ, is Planck’s constant divided by 2π. It sets the scale of quantum action and appears throughout quantum mechanics.

Planck constant: A fundamental constant equal to the quantum of action and relating a photon’s energy to its frequency. ℏ is defined by dividing this constant by 2π.
Schrödinger equation: The equation governing how a quantum state changes in time and space. ℏ controls the relation between a quantum state's time evolution and its energy.
Max Planck: German theoretical physicist whose work on blackbody radiation initiated quantum theory. His introduction of Planck’s constant supplied the quantity from which ℏ is defined.
Classical mechanics: The framework describing motion through laws such as Newton’s equations, generally without quantum effects. Classical predictions emerge when quantum effects become negligible relative to the action scale set by ℏ.
Radian: The SI unit of plane angle, defined as the angle subtended by an arc equal in length to its radius. The factor 2π arises when frequency is expressed per radian rather than per cycle.
Heisenberg uncertainty principle: A principle stating that certain pairs of observables cannot both have arbitrarily precise values in one quantum state. ℏ sets the lower bound on position–momentum uncertainty.
Arnold Sommerfeld: German theoretical physicist who expanded the old quantum theory and helped establish its mathematical notation. He introduced the reduced constant notation commonly written ℏ.
Quantum mechanics: The physical theory describing matter and radiation through states, observables, and probabilistic measurement outcomes. ℏ is a defining scale throughout this broader theory.
Angular frequency: The rate of change of phase, measured in radians per unit time. Using angular frequency pairs energy with ℏ instead of Planck’s constant.
Canonical commutation relation: The quantum relation between position and momentum operators, whose commutator is iℏ. ℏ fixes the noncommutativity of position and momentum.

Linked from 18 pages