By Professor Dr. Franz Schwabl (auth.)
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I = / dPd e -(pi+ .. +p~) = (vIJi)d . 21) This integral can also be written in spherical polar coordinates: 4 1= 1 00 dpp d- 1 ! dDde- p2 = ~! dtt~-le-t! dDd = ~F(~) ! 23) o is the gamma function. 13) 4 We denote an element of surface area on the d-dimensional unit sphere by dD d . For the calculation of the surface integral f dDd , it is not necessary to use the detailed expression for dDd. The latter may be found in E. Madelung, Die Mathematischen Hilfsmittel des Physikers, Springer, Berlin, 7th edition (1964), p.
As already mentioned, [l (E)11 is the volume of the energy shell in classical statistical mechanics. We will occasionally also refer to [l (E) as the "phase surface". We also define the volume inside the energy shell: - J D(E) dqdp h 3N N! 8(E - H(q,p)) . 5) Clearly, the following relation holds: [l (E) = dt2(E) . 1), E ::; En ::; E otherwise. 8) In the microcanonical density matrix PMC, all the energy eigenstates In} whose energy En lies in the interval [E, E + 11] contribute with equal weights.
E. there are just as many oriented parallel as anti parallel to the applied magnetic field h. ), will be discussed later in Sect. 2 . 7 ... 5 ~... -2 I!! -4 .... 1 0 Fig. 9. The entropy as a function of the energy for a two-level system (spin-~-paramagnet) ~~ Fig. 10. The temperature as a function of the energy for a two-level system (spin - ~ - paramagnet) 50 2. 6 The Canonical Ensemble In this section, the properties of a small subsystem 1 which is embedded in a large system 2, the heat bath,9 will be investigated (Fig.