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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gas in a harmonic trap ： ウィキペディア英語版 Gas in a harmonic trap The results of the quantum harmonic oscillator can be used to look at the equilibrium situation for a quantum ideal gas in a harmonic trap, which is a harmonic potential containing a large number of particles that do not interact with each other except for instantaneous thermalizing collisions. This situation is of great practical importance since many experimental studies of Bose gases are conducted in such harmonic traps. Using the results from either Maxwell–Boltzmann statistics, Bose–Einstein statistics or Fermi–Dirac statistics we use the Thomas-Fermi approximation and go to the limit of a very large trap, and express the degeneracy of the energy states ($g\_$) as a differential, and summations over states as integrals. We will then be in a position to calculate the thermodynamic properties of the gas using the partition function or the grand partition function. Only the case of massive particles will be considered, although the results can be extended to massless particles as well, much as was done in the case of the ideal gas in a box. More complete calculations will be left to separate articles, but some simple examples will be given in this article. ==Thomas Fermi approximation for the degeneracy of states== For massive particles in a harmonic well, the states of the particle are enumerated by a set of quantum numbers $()$. The energy of a particular state is given by: :$E=\backslash hbar\backslash omega\backslash left(n\_x+n\_y+n\_z+3/2\backslash right)~~~~~~~~n\_i=0,1,2,\backslash ldots$ Suppose each set of quantum numbers specify $f$ states where $f$ is the number of internal degrees of freedom of the particle that can be altered by collision. For example, a spin-1/2 particle would have $f\; =\; 2$, one for each spin state. We can think of each possible state of a particle as a point on a 3-dimensional grid of positive integers. The Thomas-Fermi approximation assumes that the quantum numbers are so large that they may be considered to be a continuum. For large values of $n$, we can estimate the number of states with energy less than or equal to $E$ from the above equation as: :$g=f\backslash ,\backslash frac=f\backslash ,\backslash frac$ which is just $f$ times the volume of the tetrahedron formed by the plane described by the energy equation and the bounding planes of the positive octant. The number of states with energy between $E$ and $E\; +\; dE$ is therefore: :$dg=\backslash frac\backslash ,fn^2\backslash ,dn=\backslash frac~\backslash frac~\backslash beta^3\; E^2\backslash ,dE$ Notice that in using this continuum approximation, we have lost the ability to characterize the low-energy states, including the ground state where $n\_\; =\; 0$. For most cases this will not be a problem, but when considering Bose–Einstein condensation, in which a large portion of the gas is in or near the ground state, we will need to recover the ability to deal with low energy states. Without using the continuum approximation, the number of particles with energy $\backslash epsilon\_$ is given by: :$N\_i\; =\; \backslash frac$ where :-1 |   for particles obeying Bose–Einstein statistics |- |$\backslash Phi=e^+1$ |   for particles obeying Fermi–Dirac statistics |} with $\backslash beta=1/kT$, with $k$ being Boltzmann's constant, $T$ being temperature, and $\backslash mu$ being the chemical potential. Using the continuum approximation, the number of particles $dN$ with energy between $E$ and $E\; +\; dE$ is now written: :$dN=\; \backslash frac$ 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「Gas in a harmonic trap」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.