I have written some code to solve and plot the time independent Schrödinger's equation with potential x^2, which has a bound state with odd integral energy eigenvalues. My code plots the graphs up to some finite x value. When I input the energy eigenvalue to be 2.9995, the wavefunction seems to increase to infinity but when I input 3.0005, it decreases to negative infinity.
I have justified this as being expected by the 'tail wag' argument - i.e. in between we expect a normalisable bound state, i.e. when E=3.
But how can I be sure that the graph will continue to increase/decrease monotonically?
Any help would be much appreciated!
If you have studied the finite square well, you know that only solutions with energy exactly equal to the eigenvalue go to $0$ at $\infty$ . For other energies they blow up exponentially (in at least one direction). This is the same phenomenon.
Instead of trying to calculate the eigenfunction at $t = 0$ by solving the equation, you could use the analytic form of the function (available in most QM texts).