Finding degeneracy of N Quantum Harmonic Oscillator

In summary, the problem involves determining the total degeneracy across N 3-D quantum harmonic oscillators, where the degree of degeneracy for a 3-D harmonic oscillator is given by (n+1)(n+2)/2 and the total energy of N 3-D quantum harmonic oscillators is given by the sum of E(i) from i=1 to 3N, or the sum of ((n(i)+0.5))*(hbar)w. The solution involves summing the degeneracy equation from n(i) i=1 to 3N and then multiplying it by the degeneracy of 1 to the third power. This method may seem unfamiliar but it is the correct approach to solving the problem
  • #1
Sekonda
207
0
Hey guys,

For a particular problem I have to determine the total degeneracy across N 3-D Quantum Harmonic oscillators.

Given that the degree of degeneracy for a 3-D harmonic oscillator is given by:

(n+1)(n+2)/2

and the Total energy of N 3d quantum harmonic oscillators is given by the sum of E(i) from i=1 to 3N, or the sum of ((n(i)+0.5))*(hbar)w.

I think I need to sum the degeneracy equation from n(i) i=1 to 3N to find the total number of degeneracies but I'm not sure what sum this reduces to!

Thanks,
Tom
 
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  • #2
Why exactly are you summing up to 3N when you have N harmonic oscillators? To get the energy of a certain state you should add the energy of all the oscialltors right. Then this is summation until N. For every one of these N things there are (n+1)(n+2)/2 different states that yield the same energy. Now you will need explicitly the energies to see in how many ways you can add N energies to get the same one. Then mulitply that by the degeneracy of 1 to the third power.

That is how I would do it at least.
 
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  • #3
Indeed, that is how I thought the problem should be worded. However the problem seems to be set out in an unfamiliar way and I'm not quite sure what is meant by the problem at all!

Though, from what you've said is what I've read around various places of the internet. Oh well, will just have to wait and see what the solution is and try and make sense of it then!

Thanks again conquest,
S
 

1. What is a Quantum Harmonic Oscillator?

A Quantum Harmonic Oscillator is a physical system that exhibits oscillatory motion due to the balance of a restoring force and a linear momentum. It is an important concept in quantum mechanics and is used to model various physical systems, such as atoms and molecules.

2. What is degeneracy in the context of a Quantum Harmonic Oscillator?

Degeneracy in a Quantum Harmonic Oscillator refers to the number of energy levels that have the same energy value. In other words, it is the number of ways in which a particular energy level can be achieved by the system.

3. How is the degeneracy of a Quantum Harmonic Oscillator calculated?

The degeneracy of a Quantum Harmonic Oscillator can be calculated using the formula: degeneracy = n + l + 1, where n is the principal quantum number and l is the angular momentum quantum number. This formula applies to a three-dimensional harmonic oscillator; for a one-dimensional oscillator, the degeneracy is simply n + 1.

4. Why is finding degeneracy important in Quantum Harmonic Oscillators?

Finding the degeneracy of a Quantum Harmonic Oscillator is important because it provides valuable information about the system's energy levels and possible states. It can also help in understanding the quantum behavior of the system and predicting its behavior in different situations.

5. Can the degeneracy of a Quantum Harmonic Oscillator be changed?

Yes, the degeneracy of a Quantum Harmonic Oscillator can be changed by altering the parameters of the system, such as the potential energy function or the mass of the oscillator. Additionally, the degeneracy can also be changed by applying external forces or perturbations to the system.

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