- #1
LagrangeEuler
- 717
- 20
For oscillator wave function
##\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.
##\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.