Complementarity in Quantum Mechanics and Classical Statistical Mechanics

Abstract

Roughly speaking, complementarity can be understood as the coexistence of multiple properties in the behavior of an object that seem to be contradictory. Although it is possible to switch among different descriptions of these properties, in principle, it is impossible to view them, at the same time, despite their simultaneous coexistence. Therefore, the consideration of all these contradictory properties is absolutely necessary to provide a complete characterization of the object. In physics, complementarity represents a basic principle of quantum theory proposed by Niels Bohr (1; 2), which is closely identified with the Copenhagen interpretation. This notion refers to effects such as the so-called wave-particle duality. In an analogous perspective as the finite character of the speed of light c implies the impossibility of a sharp separation between the notions of space and time, the finite character of the quantum of action h implies the impossibility of a sharp separation between the behavior of a quantum system and its interaction with the measuring instruments. In the early days of quantum mechanics, Bohr understood that complementarity cannot be a unique feature of quantum theories (3; 4). In fact, he suggested that the thermodynamical quantities of temperature T and energy E should be complementary in the same way as position q and momentum p in quantum mechanics. According to thermodynamics, the energy E and the temperature T can be simultaneously defined for a thermodynamic system in equilibrium. However, a complete and different viewpoint for the energy-temperature relationship is provided in the framework of classical statistical mechanics (5). Inspired on Gibbs canonical ensemble, Bohr claimed that a definite temperature T can only be attributed to the system if it is submerged into a heat bath1, in which case fluctuations of energy E are unavoidable. Conversely, a definite energy E can only be assigned when the system is put into energetic isolation, thus excluding the simultaneous determination of its temperature T. At first glance, the above reasonings are remarkably analogous to the Bohr’s arguments that support the complementary character between the coordinates q and momentum p. Dimensional analysis suggests the relevance of the following uncertainty relation (6):