Let $y_n(x)$ satisfy $y_n\prime\prime-2xy_n\prime+2ny_n=0$, where $n$ is an integer and primes denote derivatives with respect to $x$. Which plots of $\psi_n(x)=e^{-x^2/2}y_n(x)$ are acceptable?
The polynomial solutions of the Hermite equation are $y_n\propto H_n(x)$. For $n=0$, $H_0=1$, so $\psi_0$ is a Gaussian, not a constant: B is valid and A is not. For $n=1$, $H_1=2x$, so $\psi_1\propto xe^{-x^2/2}$ is odd and decays at both ends: C is valid, whereas the unbounded straight line D is not. Select B and C.
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