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Problem 4.28. Thermodynamic properties of a system of harmonic oscillators (a) Show that for one oscillator 1 f = hw + kT In(1 – e¯Bhw), (4.130) Bhw s = klBhw -Bhu In(1 1 (4.131) - e 1 (4.132) eßhw Equation (4.132) is Planck's formula for the mean energy of an oscillator at temperature T. The heat capacity is discussed in Problem 4.50.

Problem 4.28. Thermodynamic properties of a system of harmonic oscillators (a) Show that for one oscillator 1 f = hw + kT In(1 – e¯Bhw), (4.130) Bhw s = klBhw -Bhu In(1 1 (4.131) - e 1 (4.132) eßhw Equation (4.132) is Planck's formula for the mean energy of an oscillator at temperature T. The heat capacity is discussed in Problem 4.50.

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(b) Given the result (4.132), what is the mean energy of a system of N harmonic oscillators in equilibrium with a heat bath at temperature T? (c) Compare your answer with the result for the energy of N harmonic oscillators calculated in the microcanonical ensemble in Problem 4.22. Do the two ensembles give identical results? Equation (4.80) for Z is a sum over all the microstates of the system. Because the energies of different microstates may be the same, we can group together microstates with the same energy and write (4.80) as Z = £ N(EL) e-BE, (4.133) levels l where 2(Et) is the number of microstates with energy Ee. The sum in (4.133) is over all the energy levels of the system, rather than over all the microstates of the system.
Transcribed Image Text:(b) Given the result (4.132), what is the mean energy of a system of N harmonic oscillators in equilibrium with a heat bath at temperature T? (c) Compare your answer with the result for the energy of N harmonic oscillators calculated in the microcanonical ensemble in Problem 4.22. Do the two ensembles give identical results? Equation (4.80) for Z is a sum over all the microstates of the system. Because the energies of different microstates may be the same, we can group together microstates with the same energy and write (4.80) as Z = £ N(EL) e-BE, (4.133) levels l where 2(Et) is the number of microstates with energy Ee. The sum in (4.133) is over all the energy levels of the system, rather than over all the microstates of the system.
Problem 4.28. Thermodynamic properties of a system of harmonic oscillators (a) Show that for one oscillator 1 f = hw + kT In(1 – e¬Bhw), (4.130) Bhw k [ In(1 – e-9hw)], (4.131) S = eßhw – 1 1 + eßhw 1 e = hw5 (4.132) Equation (4.132) is Planck's formula for the mean energy of an oscillator at temperature T. The heat capacity is discussed in Problem 4.50.
Transcribed Image Text:Problem 4.28. Thermodynamic properties of a system of harmonic oscillators (a) Show that for one oscillator 1 f = hw + kT In(1 – e¬Bhw), (4.130) Bhw k [ In(1 – e-9hw)], (4.131) S = eßhw – 1 1 + eßhw 1 e = hw5 (4.132) Equation (4.132) is Planck's formula for the mean energy of an oscillator at temperature T. The heat capacity is discussed in Problem 4.50.
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