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##\frac{1}{\sqrt 2}(y-\frac{d}{dy})\psi_n(y)=\sqrt{n+1}\psi_{n+1}(y)##

##\frac{1}{\sqrt 2}(y+\frac{d}{dy})\psi_n(y)=\sqrt{n}\psi_{n-1}(y)##

and I interpretate ##n## as number of phonons.

Of course ##\psi_n(y)=C_ne^{-\frac{y^2}{2}}H_n(y)##.

And ##C_n=f(n)##.

Define ##\frac{1}{\sqrt 2}(y-\frac{d}{dy})=\hat{a}^+##, ##\frac{1}{\sqrt 2}(y+\frac{d}{dy})=\hat{a}##.

Why in case of infinite square well

##\psi_n(x)=\sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a}##

you can not define

##\hat{a}^+\psi_n(x) \propto \psi_{n+1}(x)##

##\hat{a}\psi_n(x) \propto \psi_{n-1}(x)##

and why quants of energy in problem of infinite square well do not have a name.

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# Phonons: For oscillator wave function

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