Consider a classical particle moving in a one-dimensional potential well u(x). The particle is in thermal equilibrium with a reservoir at temperature T, so the probabilities of its various states are determined by Boltzmann statistics.
(a) Show that the average position of the particle is given by
x =∫xe-βu(x) dx/∫e-βu(x) dx,
where each integral is over the entire x axis.
(b) If the temperature is reasonably low (but still high enough for classical mechanics to apply), the particle will spend most of its time near the bottom of the potential well. In that case we can expand u(x) in a Taylor series about the equilibrium point x0:
u(x) = u(x_0) + (x - x_0) du/dx |x_0 + ½ (x + x_0)2 d^2u/dx^2 |x_0 + 1/3! (x - x_0)^3 d^3u/dx^3 |x_0 + …
(c) If we keep the cubic term in the Taylor series as well, the integrals in the formula for x become difficult. To simplify them, assume that the cubic term is small, so its exponential can be expanded in a Taylor series (leaving the quadratic term in the exponent). Keeping only the smallest temperature-dependent term, show that in this limit x differs from x0 by a term proportional to kT. Express the coefficient of this term in terms of the coefficients of the Taylor series for u(x).
u(x) = u0[(x_0/x)^12 - 2(x_0/x)^6].
Sketch this function, and show that the minimum of the potential well is x = x0, with depth u0. For argon, x_0 = 3.9 Å and u_0 = 0.010 eV. Expand the Lennard-Jones potential in a Taylor series about the equilibrium point, and use the result of part (c) to predict the linear thermal expansion coefficient of a noble gas crystal in terms of u_0. Evaluate the result numerically for argon, and compare to the measured value α = 0.0007 K-1 (at 80 K).
The probability that the particle is in some state r is:
P_r = exp(-beta E_r)/Z
The probability that the particle is in some precise "classical state" specified by the position and momentum is obtained by putting:
E_r = u(x) + p^2/(2m)
So we have:
P[x,p] = exp[-beta u(x)]*exp[-p^2/(2m)]/Z (1)
The partition function Z makes this function properly normalized. The number of states between momenta p and p + dp and positions between x and x + dx is:
The partition function is thus:
Z =integral over x and p of dxdp/h exp[-beta u(x)]*exp[-p^2/(2m)] (2)
To obtain the probability that the particle is between positions x and x + dx, we must sum P[x,p] over all states that are compatible with such a situation. The probability that the particle s in some precise state for which the position is x and momentum is p is P[x,p]. The number of states between x and x + dx and ...
A detailed solution is given.