The question is find all the eigenvalue of the following equation
$-\frac{d^2y}{dx^2}+x^2y=\lambda y$
I have found the first function which is $y=e^{\frac{-x^2}{2}}$, however I have no clue on how to find the rest, can someone please help me with it.
$\endgroup$ 61 Answer
$\begingroup$This problem is well-known in quantum physics and goes by the name of quantum harmonic oscillator. The time-independent Schrödinger equation for this system looks like
$$\left[\frac{\hat p^2}{2m}+\frac{1}{2}kx^2\right]y=Ey$$
Here, $\hat p = -i \hbar\frac{d}{dx}$ is the momentum operator. The unknowns are the wavefunction $y$ and the energy $E$, - thus, it is an eigenvalue problem.
Setting $\hbar=1$, $m=\frac{1}{2}$, $k=2$ and renaming $E=\lambda$, we get
$$\left[\hat p^2+x^2\right]y=\lambda y$$
Or, expanding $\hat p^2$,
$$\left[-\frac{d^2}{dx^2}+x^2\right]y=\lambda y$$
which is exactly your equation.
The solutions to this problem are known; searching for the quantum harmonic oscillator you'll find tons of books, articles and lecture notes deriving the solutions.
$\endgroup$ 2
