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Quantum mechanics is most often used to describe matter on the scale of molecules, atoms, or elementary particles. However, some phenomena, particularly at low temperatures, show quantum behavior on a macroscopic scale. The best-known examples of macroscopic quantum phenomena are superfluidity and superconductivity; other examples include the quantum Hall effect and concerted proton tunneling in ice. Since 2000 there has been extensive experimental work on quantum gases, particularly Bose–Einstein Condensates. Between 1996 to 2003 four Nobel prizes were given for work related to macroscopic quantum phenomena.〔These Nobel prizes were for the discovery of super-fluidity in helium-3 (1996), for the discovery of the fractional quantum Hall effect (1998), for the demonstration of Bose–Einstein condensation (2001), and for contributions to the theory of superconductivity and superfluidity (2003).〕 Macroscopic quantum phenomena can be observed in superfluid helium and in superconductors,〔D.R. Tilley and J. Tilley, ''Superfluidity and Superconductivity'', Adam Hilger, Bristol and New York, 1990〕 but also in dilute quantum gases and in laser light. Although these media are very different, their behavior is very similar as they all show macroscopic quantum behavior. Quantum phenomena are generally classified as macroscopic when the quantum states are occupied by a large number of particles (typically Avogadro's number) or the quantum states involved are macroscopic in size (up to km size in superconducting wires). ==Consequences of the macroscopic occupation== The concept of macroscopically-occupied quantum states is introduced by Fritz London.〔Fritz London ''Superfluids'' (London, Wiley, 1954-1964)〕 In this section it will be explained what it means if the ground state is occupied by a very large number of particles. We start with the wave function of the ground state written as : with ''Ψ''₀ the amplitude and the phase. The wave function is normalized so that : The physical interpretation of the quantity : depends on the number of particles. Fig.1 represents a container with a certain number of particles with a small control volume Δ''V'' inside. We check from time to time how many particles are in the control box. We distinguish three cases: 1. There is only one particle. In this case the control volume is empty most of the time. However, there is a certain chance to find the particle in it given by Eq.(3). The chance is proportional to Δ''V''. The factor ''ΨΨ''∗ is called the chance density. 2. If the number of particles is a bit larger there are usually some particles inside the box. We can define an average, but the actual number of particles in the box has relatively large fluctuations around this average. 3. In the case of a very large number of particles there will always be a lot of particles in the small box. The number will fluctuate but the fluctuations around the average are relatively small. The average number is proportional to Δ''V'' and ''ΨΨ''∗ is now interpreted as the particle density. In quantum mechanics the particle probability flow density ''J''p (unit: particles per second per m²) can be derived from the Schrödinger equation to be :\left(\Psi (i \frac\vec -q \vec)\Psi^ * +cc \right) | style="text-align:right"|(4) |} with ''q'' the charge of the particle and the vector potential. With Eq.(1) :\left(\frac \vec \varphi - q \vec\right). | style="text-align:right"|(5) |} If the wave function is macroscopically occupied the particle probability flow density becomes a particle flow density. We introduce the fluid velocity ''v''s via the mass flow density :_s. | style="text-align:right"|(6) |} The density (mass per m³) is : so Eq.(5) results in :\left(\frac\vec\varphi-q\vec\right). | style="text-align:right"|(8) |} This important relation connects the velocity, a classical concept, of the condensate with the phase of the wave function, a quantum-mechanical concept. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「macroscopic quantum phenomena」の詳細全文を読む スポンサード リンク
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