I'm currently taking graduate classes toward my phD in physics... when I was undergraduate I learn the harmonic oscillator just solving the schrodinger equation with such potential can be derive that: E=(n+1/2)hw, the wave functions (with hermite polynomial *e^-x2). that take to pages of derivations only.(adsbygoogle = window.adsbygoogle || []).push({});

Now I'm taking QM at graduate level using the Cohen-Tannudji book, in this book the autors dedicate one chapter to the harmonic oscillator. they prove lemmas and deduce from [x,p]=ih => [a,a+]=1,...that E=(n+1/2)hw and the wave functions!

I really like that approach!, it is so elegant to get such results without using wave mechanics (solving the schrodinger equation)but

for proving that the eigenvectors are no degenerated they have to use the {|x>} representation for finding the ground state, and because the is only one solution it is not degenerated. why avoiding the {|x>} representation proving all those lemmas if you will have to used to prove the nondegeneracy of the eigenvector? Do you guys know how to prove that the eigenvalues are not degenarated without having to solve any equation?

thanks in advance

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# Learning the harmonic oscilator

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