Doubt in the quantum harmonic oscillator

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  • Thread starter Lebnm
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  • #1
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I was reviewing the harmonic oscillator with Sakurai. Using the annihilation and the creation operators ##a## and ##a^{\dagger}##, and the number operator ##N = a^{\dagger}a##, with ##N |n \rangle = n | n \rangle##, he showed that ##a | n \rangle## is an eigenstate of ##N## with eigenvalue ##n - 1##, so he concludes that ##a | n \rangle \propto | n - 1 \rangle##. But, to it be true, the spectrum of ##N## should be non-degenerated, shouldn't it? Is this true? Can I proof this?
 

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  • #2
You already label the eigen values by their energy, which relies on that the eigenvalues are non-degenerate.
 
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  • #3
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So, in the definition of the eigenstates of ##N##, I am imposing that they are non-degenerated, right? Is there some physical reason to do this? I can't see why the spectrum of ##N##, and consequently the spectrum of the hamiltonian, need to be non-degenerated in a harmonic oscillator.
 
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  • #4
DrClaude
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I can't see why the spectrum of ##N##, and consequently the spectrum of the hamiltonian, need to be non-degenerated in a harmonic oscillator.
You'll have to look in an introductory textbook, such as Griffiths.
 

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