in order to find the specific solution for the problem at hand, we must specify the appropriate boundary conditions and find the values for A, B, F, G, H and I that satisfy those conditions. Finding wavefunctions for the bound state Solutions to the Schrödinger equation must be continuous, and continuously differentiable. These requirements are boundary conditions on the differential equations previously derived, that is, the matching conditions between the solutions inside and outside the well. In this case, the finite potential well is symmetrical, so symmetry can be exploited to reduce the necessary calculations. Summarizing the previous sections: where we found , , and to be: We see that as goes to , the term goes to infinity. Likewise, as goes to , the term goes to infinity. In order for the wave function to be square integrable, we must set , and we have: and Next, we know that the overall function must be continuous and differentiable. In other words, the values of the functions and their derivatives must match up at the dividing points: These equations have two sorts of solutions, symmetric, for which and , and antisymmetric, for which and . For the symmetric case we get so taking the ratio gives Similarly for the antisymmetric case we get Recall that both and depend on the energy. What we have found is that the continuity conditions cannot be satisfied for an arbitrary value of the energy; because that's a result of the infinite potential well case. Thus, only certain energy values, which are solutions to one or either of these two equations, are allowed. Hence we find that the energy levels of the system below are discrete; the corresponding eigenfunctions are bound states. (By contrast, for the energy levels above are continuous.) The energy equations cannot be solved analytically. Nevertheless, we will see that in the symmetric case, there always exists at least one bound state, even if the well is very shallow. Graphical or numerical solutions to the energy equations are aided by rewriting them a little. If we introduce the dimensionless variables and , and note from the definitions of and that , where , the master equations read In the plot to the right, for , solutions exist where the blue semicircle intersects the purple or grey curves ( and ). Each purple or grey curve represents a possible solution, within the range