Set Up Eigenspace for Particle in 1st State: nx,ny,nz

[helpppmeee]

So what I do know about degeneracy is that it's the size of an eigenspace in a certain state. How would I go about setting up the eigenspace? Let's say for a particle in the first state with n values nx = 2, ny = 1, nz = 1

 

[bhobba]

So what I do know about degeneracy is that it's the size of an eigenspace in a certain state. How would I go about setting up the eigenspace? Let's say for a particle in the first state with n values nx = 2, ny = 1, nz = 1

All degeneracy means is that the vector space dimension of a specific outcome is greater than one.

Thanks
Bill

 

[helpppmeee]

but my prof says that there is a numerical value for each state. how would i know what it is

 

[king vitamin]

You're being extremely vague. Are you trying to find the degeneracy of each energy eigenvalue of the 3d harmonic oscillator? If so, start by finding the eigenvalues of total energy and see how this restricts nx,ny,nz.

 

[bhobba]

but my prof says that there is a numerical value for each state. how would i know what it is

You don't.

The outcome of the observation could be any element of the subspace.

Go back to the definition of observables A = ∑ yi |bi><bi|. If the yi are distinct then after an observation when the state is not destroyed (most of he time it's destroyed by the observation - its only in 'filtering' type observations it isn't) the system is in state |bi><bi|. The issue is if some of the yi are not distinct - you have degeneracy ie you can't tell what the eigenvectors are - they form a subspace. You can't tell what state its in after observation - but its only an issue if you have a filtering type observation and you would have rocks in your head setting up such a situation where you have labelled the outcomes the same. I am not of an experimental bent but I can't see how you can have such a situation in practice. Remember its only an issue if you observe it AND its not destroyed by the observation.

When you calculate the eigenvectors its purely of theoretical concern unless you observe it and that requires a specific experimental set-up.

Thanks
Bill

 

[helpppmeee]

You're being extremely vague. Are you trying to find the degeneracy of each energy eigenvalue of the 3d harmonic oscillator? If so, start by finding the eigenvalues of total energy and see how this restricts nx,ny,nz.

and what would the eigenspace be? that is my main cocern, setting up the NxN eigenspace as per my op. my prof didnt discuss how to get the values of degeneracy.

 

[helpppmeee]

for example, my prof says that the degeneracy of a 3d harmonic oscillator in the second excited state has a degeneracy of 3. The third excited state has a degeneracy of 3. The fourth excited state has a degeneracy of 1, the fifth excited state has a degeneracy of 6. I don`t understand where these values come from. My professor was very confusing about this concept.

 

[bhobba]

for example, my prof says that the degeneracy of a 3d harmonic oscillator in the second excited state has a degeneracy of 3. The third excited state has a degeneracy of 3

All that means is the eigenspace of that outcome has dimension 3. An outcome is a particular observed value - not the eigenstate the system is in if subjected to a filtering type measurement. Without that context its a total non issue.

For the Harmonic oscillator the eigenfunctions are related to the Hermite polynomials if that helps.

But I really am scratching my head about what your issue really is.

Thanks
Bill