- #1
Lebnm
- 31
- 1
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?