Use the result for fund for the two-dimensional case in problem 6.33 and compare the qualitative behavior of p(t) in one, two, and three dimensions. problem 6.60. high tempetrature limit of the ideal fermi gas if t tr at fixed density, quantum effects can be neglected and the thermal properties of an ideal fermi gas reduce to the ideal classical gas. in the follkowing we will find the first correction to the classical pressure equation of state. (a) does the presure increase or decrease when the temperature is lowered (at constant density)? that is, what is the sign of the first quantum correction to the classical pressure equation of state? the pressure is given by see (6.109) (2m)3/2 p- 3/2 de (6.264) in the high temperature limit, e 1, we can make the expansion. (6.265a) ele-+ 1+-e-) e-1e-e-p) (6.265b) if ue use (6.265b) we ohtain /2e(1-ee)dr- (6.266) use (6.266) to show that p is given by m3/2(kt)/2 p 2/23/213 (6.267) 25/2 chapter 6. many-particle systems 351 (b) derive an expression for n similar to (6.267). eliminate a and show that the leading order correction to the equation of state is given by - nur1+ 8/2 ph pv (6.268a) 4 (mkt/2 (6.268b) (e) what is the condition for the carrection term in (6.268h) to be small? note that as the temperature is lowered at constant density, the pressure increases. what do you think would be the effect of base statistics in this context (see problem 6.61)? mullin and blaylock (2003) have emphasized that it is misleading to interpret the siga of the correction term in (6.268b) in terms of an effective repulsive exchange "force," and stress that the positive sign is a consequence of the symmetrization requirement for same spin fermions. problem 6.61. high temperature limit of ideal bose gas if t t at fixed density, quantum effects can be neglected and thbe thermal properties of an ideal bose gas reduce to those of the ideal classical gas. does the pressure increase or decrease when the temperature is lowered (at constant density)? that is, what is the first quantum correction to the classical equation of state? the pressure is given by see (6.109) 21/2m/2(t/2 /dz 32h8 p- (6.269) e-p-1 as in problem 6.60 and show that follow the same procedure /2 ph3 nat[ pv- (6.270) 2 (mkt/2 we see that ss the temperature is lowered at constant density, the pressure becomes less than its classical valne