The potential energy of a one-dimensional, anharmonic oscillator may be written as
V(q)=cq2−gq3−fq4,
where c, g and f are positive constants; quite generally, g and f may be assumed to be very small in value. Show that the leading contribution of anharmonic terms to the heat capacity of the oscillator, assumed classical, is given by
23k2(c2f+45c3g2)T
and, to the same order, the mean value of the position coordinate q is given by
43c2gkT.