Physics for Scientists and Engineers, Vol. 1
6th Edition
ISBN: 9781429201322
Author: Paul A. Tipler, Gene Mosca
Publisher: Macmillan Higher Education
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Chapter 35, Problem 10P
To determine
The expectation value of
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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. This result corresponds to the classical limit and is
shown to be an example of the equipartition theorem. In this limit the
energy kT is large in comparison to ħw, the separation between
energy levels. Hint: expand the exponential function
1
ē = ħw
+
eBhw
(c) Show that at low temperatures for which ħw> kT ,
E/N = hw(+e-Bhw)
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).
Find the probabilities that a particle can be found between x = 0.45 and 0.55 when the particle in limited to the x axis and has wave function.
ψ = {ax, 0 ≤ x ≤ 1 ; 0, x< 0 and x >1.
Also find the average (expectation value) of x.
Consider a particle with the following wave-function:
,xL
and L and A are constants.
(a) What is the normalization constant A (in terms of L)?
(b) What is (in terms of L)? ? (ie, )
(c) What is the probability that the particle will be found within x=0 and L/2?
Chapter 35 Solutions
Physics for Scientists and Engineers, Vol. 1
Ch. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7PCh. 35 - Prob. 8PCh. 35 - Prob. 9PCh. 35 - Prob. 10P
Ch. 35 - Prob. 11PCh. 35 - Prob. 12PCh. 35 - Prob. 13PCh. 35 - Prob. 14PCh. 35 - Prob. 15PCh. 35 - Prob. 16PCh. 35 - Prob. 17PCh. 35 - Prob. 18PCh. 35 - Prob. 19PCh. 35 - Prob. 20PCh. 35 - Prob. 21PCh. 35 - Prob. 22PCh. 35 - Prob. 23PCh. 35 - Prob. 24PCh. 35 - Prob. 25PCh. 35 - Prob. 26PCh. 35 - Prob. 27PCh. 35 - Prob. 28PCh. 35 - Prob. 29PCh. 35 - Prob. 30PCh. 35 - Prob. 31PCh. 35 - Prob. 32PCh. 35 - Prob. 33PCh. 35 - Prob. 34PCh. 35 - Prob. 35PCh. 35 - Prob. 36PCh. 35 - Prob. 37PCh. 35 - Prob. 38P
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