Quantum Well with Quadratic Potential

@Karan Chatrath has explained how to tackle this problem conceptually. I like to make a plot of $\Psi(\pi)$ vs. $E$ . The energies at which the graph crosses the horizontal axis $(\Psi(\pi) = 0)$ are the allowed energies, given initial conditions (for each $E$ ) of $\Psi(-\pi) = 0$ and $\frac{d}{dx} \Psi(-\pi) = 1$ .

Since this problem was formulated in terms of the time-independent $SE$ , there is no time-evolution of the wave function. The wave function for the lowest energy state $E_1$ is symmetric about $x = 0$ and it has a single peak at $x = 0$ , meaning that the particle is most likely to be found near the origin, where the potential is minimized. This seems fairly analogous to the low-energy classical picture. A low-energy classical oscillator has to remain near the location of minimum potential. At high energy, a classical oscillator moves back and forth over a wide range, and is not confined to a particular location. With high $E$ , the quantum wave function has many peaks spread throughout the well.

I have plotted $\Psi^2(x)$ for a high value of $E$ $(E = 93.56 )$ . You can see that the probability density is more uniform. On a large scale, there is no region of the well that is strongly preferred over any other. However, there is a slight preference for the high-potential regions. I think this is because a classical oscillator has low speed at high potential, and high speed at low potential. Therefore, the classical oscillator spends more of its time in the high potential regions.

I am reading up how to solve the given differential equation analytically as closed form solutions are available, but to solve this problem, I relied on numerical methods.

$\frac{d^2 \Psi}{dx^2} = (x^2-E)\Psi$ $\Psi(0) = 0\ ; \ \frac{d\Psi}{dx}(0) =1$

The initial derivative of $\Psi(x)$ can be taken as anything. The solution is simply scaled in proportion to the initial derivative. $E$ is varied in small increments and the allowable energies are those for which:

$\lvert \Psi(\pi)\rvert < \epsilon$ Where $\epsilon$ is a small numeric tolerance like $10^{-3}$ , for example. Essentially, we search for the solutions which satisfy the non-imposed boundary condition $\Psi(\pi)=0$ . The following plot illustrates the first five allowable solutions for the quantum harmonic oscillator.

I also do not know how to interpret when the quantum solution starts to agree with the classical solution. How would I visualize such an agreement? In classical oscillators, there isn't any notion of a 'wave function'.