195(b) The root-mean-square fluctuation is(c) When kT << hw,When kT >> hw,
(E) -+ kT, AE -+ kT.
2033
Consider a system of No non-interacting quantum mechanical oscilla-
tors in equilibrium at temperature T. The energy levels of a single oscillator
are
Em = (rn + 1/2)7/V with rn = 0, 1,Z... etc.
(7 is a constant, the oscillators and volume V are one dimensional.)
(a) Find U and C, as functions of T.
(b) Sketch U(T) and C,(T).
(c) Determine the equation of state for the system.
(d) What is the fraction of particles in the m-th level?
(SUNY, Bufulo)
Solution:
(a) The partition function is
The internal energy is
a coth - 7P
ap 2v 2v
U = -No - In z =