al harmonic oscilla a heat bath
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Transcribed Image Text:One-dimensional harmonic oscillators in
equilibrium with a heat bath
(a) Calculate the specific heat of the one-dimensional
harmonic oscillator as a function of temperature
(b) Plot the T -dependence of the mean energy per
particle E/N and the specific heat c. Show that E/N →
kT at high temperatures for which kT > hw
In this limit the energy kT is large in comparison to hw
, the separation between energy levels. Hint: expand
the exponential function
1
ē = ħw
+
eBhw
Transcribed Image Text:(c) Show that at low temperatures for which ħu> kT ,
E|N :
= ħw(}+e-8hw)
What is the value of the heat capacity?
Why is the latter so much smaller than it is in the high
temperature limit?
Why is this behavior different from that of a two-state
system?
(d) Verify that S>0 as T→O in agreement with the
third law of thermodynamics, and that at high T,
S> kN In(kT /hw).
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