# referencing: 3-6. Use either the expression for (E) in Example 3-1 or the one in Problem…

referencing:

3-6. Use either the expression for (E) in Example 3-1 or the one in Problem 3-5 to show that hy B and that (E)-_ 0 T_00 as Postulate That the Average Ensemble Energy Is Equal to the Observed Energy of a System ls EXAMPLE 3-1 uation for (E) for the simple system of a (bare) proton in a magnetic eld B ON: The energy of a bare proton in a magnetic field B, is given by Ey y is a quantity called the magnetogyric ratio. The partition function of _Ihy8, where consists of just two terms The average energy is obtained from either Equation 3.20 or 3.21: a In Q This expression for (E) (in units of hy B,/2) is plotted against T (in units of hy B./2kg in Figure 3.2. Note that (E) _-hyBz/2 as T _ 0 and that (E) _ 0 as T _ oo. As T0, there is no thermal energy, so the proton orients itself parallel to the magnetic field with certainty. As T _ oo, however, the thermal energy of the proton increases to such an extent that the proton is equally likely to point in either direction. TI (hyB,/2kB) 25 50 iy restriction hples of bosons are alpha lIlS out that fermions and bosons constitute o the summation in Equation 3.37 for the case of fermions. n nature. We must recognize the occupancy requirements of all the known einback now to the summation in when we attempt to carry out the summation in Equation 3.37. and bosons when w thes cal fermions can occupy the same single-particle energy st Let’s go back now cause no two identical ecauin which two or more es inrefore, the indice two identi e indices are the same cannot be included in the sum- s i, j, k, are not independent of one another, so of Equation 3.37 poses problems for the s mation V, T) by means direct evaluation of Q(N, V fermions EXAMPLE 3-5 with energies _ , in Equation 3.37. system of two noninteracting identical fermions, each of which has states e,, _3, and _4. Enumerate the allowed total energies in the summation SOLUTION: For this system i,j-1 of the 16 terms that would occur in an unrestricted evaluation of Q, only six are allowed for two identical fermions; these are the terms with energies

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