Ok, im reading this part in my book that tries to explain the reason for energy quantisation. It first shows that the time independent S-eq is a an eigenvalue equation, and writes it as: d^2/dx^2 Y(x) = 2m/h^2 [V(x) - E(x)] Y(x) where i use Y instead of "Psi". It gives a graph of an arbitrary potential approaching V- as x-> -inf, V+ as x-> +inf and has a min of Vmin. It considers 4 cases: E < Vmin, Vmin< E< V-, V-< E < V+, E > V+. I understand the first case is physically unacceptable because Y(x) has to be concave up if at some x=x* Y(x*)>0, concave down if Y(x*)<0, and "escapes" from the x-axis if Y(x*)=0 and due to the probabilistic nature of the wavefunction, Y(x) must be finite. Now for the second case, Vmin < E < V-, there are two points of intersection between E and V, call them x1 for the left intersection and x2 for the right. I understand the Y(x) must oscillate when x is between x1 and x2. Also if x<x1, then it can converge either to the left or to the right and if x>x2 then also it can converge either to the left or to the right. Now since we're assuming V is finite and continuous everywhere it follows that d^2Y(x)/dx^2 and dY(x) are also continuous. The book tries to explain that because the d^2Y(x)/dx^2 depends on E, then the curvature of Y(x) depends on E, then there may be certain discrete values of E for which the solution in the internal region will join smoothly with to the solutions in the external regions, and they say that such solutions which are finite, continuous and having a continuous derivative everywhere are eigenfunctions of the time independent S-eq. I dont really understand what they mean...why cant you have any value of E creating this smooth connection between the curves?