Obtain expressions for the thermal energy and the phonon heat capacity of (a) a system of two ha

Obtain expressions for the thermal energy and the phonon...

Key Concepts

Partition Function

The partition function is a fundamental concept in statistical mechanics that encapsulates all the statistical properties of a thermodynamic system. It is the sum over all energy states weighted by the Boltzmann factor. For both the harmonic oscillator and the two-level system, the partition function is used to derive expressions for thermal energy and heat capacity by calculating appropriate derivatives with respect to temperature.

Heat Capacity

Heat capacity is a measure of how much the internal energy of a system changes with temperature. In the context of the two models, the phonon heat capacity derived from harmonic oscillators reflects the cumulative effect of many vibrational modes leading to a continuous temperature dependence, whereas in the two-level system, the heat capacity can exhibit sharp features such as peaks when the thermal energy is comparable to the energy difference between the two states.

Harmonic Oscillator

A harmonic oscillator in statistical physics is a system whose energy levels are equally spaced and quantized. In the context of thermal energy and heat capacity, mathematical tools such as the partition function are used to sum over these energy levels to obtain expressions for average energy and its temperature derivative. This model is especially important in describing vibrational modes in solids, where phonons behave like quantized harmonic oscillators.

Two-Level System

A two-level system is a simplified model that considers only two possible energy states, typically a ground state and an excited state. Due to the discrete nature of its energy spectrum, the statistical properties, such as the partition function, thermal energy, and heat capacity, are computed with just two terms, which leads to a very different temperature dependence compared to systems with many equally spaced energy levels.

Transcript

00:01 So this question asks about the difference between the harmonic oscillator and an harmonic oscillator, and ask whether the energy is equally spaced between the energy levels.

00:20 So let's first look at the simple harmonic oscillator.

00:25 And before we start, i would mention you can actually find the concepts of the knowledge in page 392 to page 393.

00:41 Okay, let's start with a harmonic oscillator.

00:45 So in a homolic oscillator system, the energy of the vibration equals to what? equals to a vibration quantum number v plus one half times blunt constant.

01:10 And times the mu, which is a frequency of the vibration.

01:16 And this we call the quantum number.

01:28 To see whether the energy of the states is equally spaced, we can just say, right, we can set a quantum number v equals to a.

01:41 A is any integer, right? and the next states, the quantum number, would be a plus 1.

02:02 And the energy difference between the state and the next states, right, is ev prime minus ev.

02:16 Actually, e equals to, so ev prime equals to v equals to a plus 1, right? which is a quantum number and plus one half times hmew and minus the ev, right? which here is actually a, which is a quantum number plus one half times hv.

02:54 And it actually equals to, if you do, hmu, right? so we can see for harmonic asset, given any states, and the difference between these states and previous states, or the next states is always hmu, so the energy of the states is actually equally spaced for the harmonic oscillator.

03:26 And now we can try to solve for the unharmonic oscillator, since i run out the space of the whiteboard, i have to wipe out them first.

03:54 So for an harmonic oscillator, if you check the book, you will know the energy of the vibration is a little bit different from the harmonic oscillator case, which equals to same.

04:13 We have a quantum number plus one -half times h -mew, but there is another part, which is the quantum number, plus one half, but it's square and times hmu and xe...

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