# Free eneryg of a harmonc oscillator

• stunner5000pt
In summary, we discussed the expression for the free energy, entropy, and heat capacity for a one-dimensional harmonic oscillator and how to calculate them using the partition function.
stunner5000pt
tau = Boltzmann constant times absolute temperature
A one dimensional harmonic oscillator has an infinite series of equally spaced energy states, with $\epsilon_{s} = s \hbar \omega$ where s is a positive integer or zero and omega is the calssical frequency of the oscillator.

a) Find the free energy of the system

b) Find the entropy an Heat capacity

well ok the parititon function here is

$$Z = \sum_{s=0}^{\infty} \exp(\epsilon_{s}/ \tau) = 1 + e^{\frac{\hbar \omega}{\tau}} + e^{\frac{2\hbar \omega}{\tau}} + e^{3\frac{\hbar \omega}{\tau}} + ... = \frac{e^{\frac{\hbar \omega}{\tau}}}{e^{\frac{\hbar \omega}{\tau}} -1}$$

so to find the free energy i simply have to find the natural log of Z and multiply that by -tau, which becomes
$$-\tau \left(\frac{\hbar \omega}{\tau} - \log (e^{\frac{\hbar \omega}{\tau}} -1 )\right)$$

can this simplify furhter though?

b) to find the netropy we simply find the partial derivative of the free energy wrt tau while holding V contsnat and we get the negative entrpy

for the specific heat capacaity i am not usre...
isnt $C_{V} = \frac{3}{2}$ since N = 1 as there is only one one dimension??

thank you for the help!

Hello,

I can help you with your questions. Let's start with finding the free energy of the system. You are correct in your calculation, and the expression for the free energy can be simplified further by using the identity ln(x-1) = -ln(1-x). This will give you:

F = -\tau \left(\frac{\hbar \omega}{\tau} + \log (1-e^{-\frac{\hbar \omega}{\tau}} )\right)

Moving on to finding the entropy, you are correct that it can be found by taking the partial derivative of the free energy with respect to temperature (tau). The resulting expression will be:

S = \frac{\hbar \omega}{\tau} + \frac{\hbar \omega}{e^{\frac{\hbar \omega}{\tau}} -1}

Lastly, for the heat capacity, you are correct that it can be found by taking the partial derivative of the energy (which is the same as the free energy in this case) with respect to temperature. However, the expression for the heat capacity will depend on the dimensionality of the system. In this case, for a one-dimensional system, the heat capacity will be:

C_V = \frac{1}{2} k_B \left(\frac{\hbar \omega}{k_B T}\right)^2 \frac{e^{\frac{\hbar \omega}{k_B T}}}{(e^{\frac{\hbar \omega}{k_B T}} - 1)^2}

Where k_B is the Boltzmann constant.

I hope this helps. Let me know if you have any further questions.

## 1. What is the definition of free energy in a harmonic oscillator?

The free energy of a harmonic oscillator is a measure of the energy that is available to do work. It takes into account both the potential energy and the kinetic energy of the oscillator.

## 2. How is the free energy of a harmonic oscillator calculated?

The free energy of a harmonic oscillator can be calculated using the equation:
F = E - TS
where F is the free energy, E is the total energy, T is the temperature, and S is the entropy of the system.

## 3. What factors affect the free energy of a harmonic oscillator?

The free energy of a harmonic oscillator is affected by the temperature of the system, the amplitude of the oscillator, and the frequency of oscillation.

## 4. How does the free energy of a harmonic oscillator change with temperature?

The free energy of a harmonic oscillator is directly proportional to temperature. As temperature increases, the free energy also increases, indicating that more energy is available to do work.

## 5. What is the significance of the free energy of a harmonic oscillator in thermodynamics?

The free energy of a harmonic oscillator is an important concept in thermodynamics as it represents the maximum amount of work that can be extracted from the system. It is also used to determine the stability of a system, with lower free energy indicating a more stable system.

Replies
5
Views
177
Replies
4
Views
1K
Replies
1
Views
1K
Replies
0
Views
126
Replies
9
Views
1K
Replies
11
Views
2K
Replies
3
Views
432
Replies
1
Views
2K
Replies
1
Views
1K
Replies
4
Views
2K