Incommensurability (physics)
In physics and metrology , one speaks of incommensurability (negative form of Latin "measurable together") of physical quantities or measured quantities if these quantities are "not comparable with one another" according to various criteria.
measuring technology
In general, two quantities are called commensurable if they can be measured with the same degree or if they are comparable. Two quantities are incommensurable if they cannot be measured or compared.
For example, the stiffness of a solid cannot be compared with that of a gas, because a gas does not have the property of stiffness at all. Gases and solids are incommensurable with regard to this property.
Incommensurable are all physical terms that do not agree with regard to their physical dimension (i.e. their measure). Here the conceptual content of "incommensurable", i.e. H. “Not to be measured by the same measure”, particularly clear: length (in meters) cannot be compared with weight (in kilograms). If two quantities have the same physical dimension, but different units, they are considered commensurable. For example, a length in the unit “mile” or “meter” can be converted into the other system of measurement so that the sizes can then be directly compared.
International system of units
In the international system of units SI , all units are traced back to combinations (power products) of only 7 base units and 22 (also combined from the base units) "units with special names" .
Very different and incommensurable sizes can still have the same unit. So have z. B. the two quantities heat capacity and entropy have the same unit Joule per Kelvin . In the same way, the mechanical work and the torque share the unit Newton meter . In some cases, differentiating between different quantities with the same units is made easier by assigning different unit names to the same units. So z. For example, the unit “1 / s” is called “Hertz” , “Becquerel” or “Radiant per second” , depending on whether the designated variable is “cycle per second” , “decay events per second” or “angular velocity” .
Period relationships
The terms commensurable and incommensurable are used in physics as well as in mathematics ( incommensurability of numbers ) to express the fundamental difference between rational and irrational numbers in the relationships between the period lengths (or wavelengths or frequencies). Period relationships described by rational numbers are called commensurable and lead to periodic spatial patterns or movements, irrational period relationships are called incommensurable and correspond to quasi-periodic or chaotic structures.
Commensurable and incommensurable superstructures occur e.g. B. in solids , especially if two competing interactions in a crystal lattice or in a magnet favor different period lengths (see e.g. ANNNI and Frenkel-Kontorowa model ).
Quantum mechanics
In quantum mechanics , two quantities or operators are called incommensurable if their commutator is not equal to 0. The terms compatible, compatible, commuting or commensurable are used as an opposing word. Incommensurable quantities cannot be measured sharply at the same time, this describes the Heisenberg uncertainty principle .