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.
Related questions
Question
![**3. Kittel Ch3-3. Free energy of a harmonic oscillator.**
A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with \( \varepsilon_s = s\hbar\omega \), where \( s \) is a positive integer or zero, and \( \omega \) is 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
\[
F = \tau \log[1 - \exp(-\hbar\omega / \tau)]
\]
(87)
Note that at high temperature such that \( \tau \gg \hbar\omega \), we may expand the argument of the logarithm to obtain \( F \approx \tau \log(\hbar\omega / \tau) \).
(b) From (87) show that the entropy is
\[
\sigma = \frac{\hbar \omega / \tau}{\exp(\hbar \omega / \tau) - 1} - \log[1 - \exp(-\hbar\omega / \tau)]
\]
(88)
The entropy is shown in Figure 3.13 and the heat capacity in Figure 3.14.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d2fdd51-a813-4b36-89e9-f9581acfc2ee%2F191c9304-1f46-40f1-a91d-5aad47e93b4c%2Fhx6ne4l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**3. Kittel Ch3-3. Free energy of a harmonic oscillator.**
A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with \( \varepsilon_s = s\hbar\omega \), where \( s \) is a positive integer or zero, and \( \omega \) is 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
\[
F = \tau \log[1 - \exp(-\hbar\omega / \tau)]
\]
(87)
Note that at high temperature such that \( \tau \gg \hbar\omega \), we may expand the argument of the logarithm to obtain \( F \approx \tau \log(\hbar\omega / \tau) \).
(b) From (87) show that the entropy is
\[
\sigma = \frac{\hbar \omega / \tau}{\exp(\hbar \omega / \tau) - 1} - \log[1 - \exp(-\hbar\omega / \tau)]
\]
(88)
The entropy is shown in Figure 3.13 and the heat capacity in Figure 3.14.
Expert Solution
Step 1
We have given a energy of one dimension hormonic oscillator.
We can write the partition function for this system and further solve as in step 2.
Step by step
Solved in 3 steps with 2 images
