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While I acknowledge that this topic has been discussed extensively, and I've read numerous similar questions along with their respective answers, I am still struggling to comprehend why all the eigenvectors of the harmonic oscillator are non-degenerate.

My confusion stems from a particular proof presented in the book "QUANTUM MECHANICS Volume I Basic Concepts, Tools, and Applications" by Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, specifically in Chapter five on page 510.

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I grasp that the proof establishes the non-degeneracy of $|\phi_n\rangle$, and subsequently, proving that $|\phi_{n+1}\rangle$ is proportional to it implies its non-degeneracy. If this were my sole perspective, everything would make sense. However, I'm struggling to understand why having all vectors proportional to $|\phi_n\rangle$ doesn't lead to a problem of degeneracy.

In a nutshell why proportionality does not lead to degeneracy?

These are some of the questions that I have read

Degeneracy of the ground state of harmonic oscillator with non-zero spin

Proof that the one-dimensional simple harmonic oscillator is non-degenerate?

my2cts
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Manuel Borra
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2 Answers2

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The idea is to identify all the distinct eigenstates $|\phi_n\rangle$ of the number operator $\hat{n}$. These eigenstates are normalized $\langle\phi_n|\phi_n\rangle=1$. In the process, we find that, given an eigenstate $|\phi_n\rangle$, the state $\hat{a}^{\dagger}|\phi_n\rangle$ is not the same state as $|\phi_n\rangle$. In fact, we find that it is proportional to $|\phi_{n+1}\rangle$. There may seem to be several different candidates for $|\phi_{n+1}\rangle$ all associated with the eigenvalue $(1+n)$, but they are all proportional to $\hat{a}^{\dagger}|\phi_n\rangle$. Now clearly, all the states that differ only by a constant proportionality factor must be identically the same state. All that remains is to normalize these states to remove the proportionality constants, as required for the eigenstates. Then they would all be equal and there would not be any degeneracy. Does that make sense?

flippiefanus
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Since any states are proportional they are identical after normalisation. Hence there is only one state and there is no degeneracy.

my2cts
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