Degeneracy of the Quantum Linear Oscillator

In summary, while reading about the quantum linear oscillator in "Classical and Statistical Thermodynamics" by Ashley H. Carter for my statistical mechanics homework, I came across a point about the degeneracy of the system. It states that the energies are equally spaced and the ground state has a zero-point energy. However, the book does not provide further explanation about why the degeneracy is one for all states. The problem is mentioned in chapter 15, "The Heat Capacity of a Diatomic Gas" and it is described as an assembly of N one-dimensional harmonic oscillators that are loosely coupled and free to vibrate in one dimension. This system is nondegenerate because the vibration of a diatomic is modeled as a one-dimensional harmonic
  • #1
mavanhel
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So, today while doing my homework for statistical mechanics I was reading about the quantum linear oscillator in the textbook, "Classical and Statistical Thermodynamics" by Ashley H. Carter. In it, after discussing the quantized energy it says:

"Note that the energies are equally spaced and that the ground state has a 'zero-point' energy equal to [tex]\frac{hv}{2}[/tex]. The states are nondegenerate in that [tex]g_{j} = 1[/tex] for all [tex]j[/tex]."

The book gives no further clarification on this point as, but I was wondering why the degeneracy for this problem is one for all states.

This problem occurs at the beginning of chapter 15, "The Heat Capacity of a Diatomic Gas" and makes the following considerations:
  • consider an assemply of N one-dimensional harmonic oscillators.
  • The oscillators are loosely coupled (ie. small energy exchange between them).
  • The oscillators are free to vibrate in one dimension freely.

This does not tell us anything about the degeneracy, so why is this system nondegenerate?
 
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  • #2
The vibration of a diatomic is modeled as a one-dimensional harmonic oscillator, for which there is a single eigenfunction for each eigenenergy. There is nothing more to it.
 

1. What is the degeneracy of the quantum linear oscillator?

The degeneracy of the quantum linear oscillator refers to the number of energy levels with the same energy value. In other words, it is the number of different quantum states that have the same energy. This is a characteristic of quantum systems and is related to the symmetry of the system.

2. How does the degeneracy of the quantum linear oscillator affect its behavior?

The degeneracy of the quantum linear oscillator does not have a direct effect on its behavior, but it does play a role in determining the probability of finding the oscillator in a particular energy state. Higher degeneracy means there are more possible energy states for the oscillator to occupy, which can affect the overall distribution of energy levels.

3. Can the degeneracy of the quantum linear oscillator be changed?

The degeneracy of the quantum linear oscillator is a characteristic of the system and cannot be changed. However, it can be affected by external factors such as changes in the shape or symmetry of the system, which can alter the energy levels and thus the degeneracy.

4. How is the degeneracy of the quantum linear oscillator calculated?

The degeneracy of the quantum linear oscillator can be calculated using mathematical equations that take into account the symmetry of the system. The exact calculation method may vary depending on the specific system and its properties.

5. What are some real-world applications of the degeneracy of the quantum linear oscillator?

The degeneracy of the quantum linear oscillator is a fundamental concept in quantum mechanics and has many practical applications. It is used in the study of molecular and atomic structures, as well as in the development of quantum technologies such as quantum computers and sensors. The degeneracy also plays a role in understanding the behavior of materials and their properties at the quantum level.

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