kvantisointiasetus

Kvantisointiasetus, or quantization condition, is a principle in physics stating that certain physical quantities can take only discrete values in a system because of wave properties or boundary constraints. It arose in the early quantum theory to explain why many observables are not continuous but come in distinct steps.

One classic form is the Bohr–Sommerfeld quantization condition, written as the integral ∮ p dq = 2πħ n, where p is the conjugate momentum, q is the coordinate, ħ is the reduced Planck constant, and n is an integer. This semiclassical requirement selects allowed orbital motions and energy levels in systems with bound states. In modern quantum mechanics, discrete values emerge from solving the Schrödinger equation with appropriate boundary conditions, producing a discrete spectrum of eigenvalues for bound states.

Common examples include the particle in a one-dimensional box, which yields energy levels proportional to n^2, and the quantization of angular momentum, where the squared angular momentum L^2 has eigenvalues ħ^2 l(l+1) with integer l. In hydrogen-like atoms, energy levels depend on the principal quantum number n, reflecting underlying quantization of the Coulomb problem. More generally, quantization conditions appear when wavefunctions must be single-valued and satisfy standing-wave boundary conditions, leading to discrete allowed states.

In addition to atomic physics, quantization conditions appear in broader contexts such as topological and condensed-matter phenomena, including Dirac's quantization condition for magnetic charge and the quantized conductance in the quantum Hall effect. Kvantisointiasetus thus links classical action concepts to the discrete spectra that characterize quantum systems.