Asymptotic behavior and the Hermite Polynomial

$$\eta \gg K : \ \psi_{as}(\eta)=Ae^{-\eta^2/2} \ \Rightarrow \psi(\eta)=Af(\eta)e^{-\eta^2/2}$$

$$\frac{d^2\psi}{d \eta^2} = \left(\frac{d^2 f(\eta)}{d\eta^2}-2\eta \frac{df(\eta)}{d\eta}+(\eta^2-1)f(\eta)\right)e^{-\eta^2/2}$$

$$\frac{d^2f(\eta)}{d\eta^2}-2\eta\frac{df(\eta)}{d\eta} + (K-1)f(\eta) = 0$$ (2.2)

We introduce the ansatz, based on the middle term and the overall structure of Equation 2.2, that $f(\eta)$ will have a derivative-cyclic solution (such as the Legendre and Laguerre functions) with the kernel $\exp(-\eta^2)$. Indeed the first two derivatives suggest a correct direction,

$$\frac{d}{d\eta}e^{-\eta^2} = -2\eta e^{-\eta^2}, \qquad \frac{d^2}{d\eta^2}e^{-\eta^2} = -2e^{-\eta^2} + 4\eta^2 e^{-\eta^2}$$ (2.3)

In accordance with the Laguerre polynomial solution (specifically the Rodriguez formulation), we would try the following solution:

$$f(\eta)=(-1)^n \alpha e^{\eta^2}\frac{d^n}{d\eta^n}e^{-\eta^2}$$ (2.4)

We sample the $n=2$ case,

$$f_2(\eta)=\alpha(-2+4\eta^2), \ f'_2(\eta)=\alpha8\eta, \ f''_2(\eta)=8\alpha \ \rightarrow \ 8 -2\eta(8\eta) + (K-1)(4\eta^2-2)=0$$

This equation is satisfied when $K-1=4$ or generally $K=2n+1$. This is the quantization condition. The reader might not be satisfied with our methodology, and so we wish to re-demonstrate this condition with another approach. Here we suppose a series solution to $f(\eta)$.

$$f(\eta)=\sum^{\infty}_{j=0}a_j\eta^j \ \longrightarrow \sum^{\infty}_{j=0}\left((j+1)(j+2)a_{j+2}-2ja_j + (K-1)a_j\right)\eta^j =0$$

$$a_{j+2}=\frac{2j+1-K}{(j+1)(j+2)}a_j$$ (2.5)

As $j\rightarrow \infty$ the series will approach zero at a rate of $1/j$ which is a condition for divergence. We thus need the series to terminate for some $j=n \ \ni \ K=2n+1$ so that $a_{n+1}=0$. From our definition of K, we have found the energy levels to be

$$E=(2n+1)\frac{\hbar \omega}{2}$$ (2.6)