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 hw, the separation between energy levels. Hint: expand the exponential function 1 ē = hw - + eBhw (c) Show that at low temperatures for which hw> kT , E/N = hw(}+e-Phuw) 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→0 in agreement with the third law of thermodynamics, and that at high T,S→ kN In(kT / Fw).

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 hw, the separation between energy levels. Hint: expand the exponential function 1 ē = hw - + eBhw (c) Show that at low temperatures for which hw> kT , E/N = hw(}+e-Phuw) 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→0 in agreement with the third law of thermodynamics, and that at high T,S→ kN In(kT / Fw).

University Physics Volume 3

17th Edition

ISBN:9781938168185

Author:William Moebs, Jeff Sanny

Publisher:William Moebs, Jeff Sanny

Chapter7: Quantum Mechanics

Section: Chapter Questions

Problem 90CP: In STM, an elevation of the tip above the surface being scanned can be determined with a great...

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