3. Kittel Ch3-3. Free energy of a harmonic oscillator. A one-dimentional harmonic oscillator has an infinite series of series of equally spaced energy states, with &=sha, where s is a positive integer or zero, and wis the classical frequency of the oscillator. We have chosen the zero energy at the state s-0. (a) Show that for a harmonic oscillator the free energy is F = Tlog[1-exp(-ħw/T)] Note that at high temperature such that 7 >> ħw we may expand the argument of the logarithm to obtain F=Tlog(ħw/T). (b) From (87) show that the entropy is O= hω/τ exp(ħw/T)-1 (87) -log[1-exp(-ħw/T)] (88) The entropy is shown in Figure 3.13 and the heat capacity in Figure 3.14.

3. Kittel Ch3-3. Free energy of a harmonic oscillator. A one-dimentional harmonic oscillator has an infinite series of series of equally spaced energy states, with &=sha, where s is a positive integer or zero, and wis the classical frequency of the oscillator. We have chosen the zero energy at the state s-0. (a) Show that for a harmonic oscillator the free energy is F = Tlog[1-exp(-ħw/T)] Note that at high temperature such that 7 >> ħw we may expand the argument of the logarithm to obtain F=Tlog(ħw/T). (b) From (87) show that the entropy is O= hω/τ exp(ħw/T)-1 (87) -log[1-exp(-ħw/T)] (88) The entropy is shown in Figure 3.13 and the heat capacity in Figure 3.14.