# Energy levels splitting in double-well potentials

As it is well known, the tunnel splitting (which are certain differences between energy levels) is the characteristic of the energy spectrum for the double-well potentials. When we calculate the energy eigenvalues for the ground state and the other higher states, we see that the energy levels satisfy the inequality $E_1-E_0 < E_3-E_2 < E_5-E_4 < …$ which implies that the adjacent energy levels are paired together. For example, $E_1$ is close to $E_0$ but far from $E_2$. Similarly, $E_2$ is far from $E_1$ but close to $E_3$. My question is that why we observe such a manner in double-well potentials? In other words, what is the physical interpretation of this effect?

The lowest energy level corresponds to a symmetric wave function $$\psi_0(x)$$ while the second is an antisymmetric wave function $$\psi_1(x)$$. Their squared amplitudes are almost equal $$|\psi_0(x)|^2\approx |\psi_1(x)|^2$$ and both correspond to a particle equally likely to be found in the left or right well (to get a definite well one needs to be in the state $$(\psi_0(x)\pm \psi_1(x))/\sqrt{2}$$). As one increases the quantum number one gets similar pairs of nearly identical wave functions, only differing by having a zero between the wells. Inside each well (especially if one looks at superpositions that are nearly zero in the other well) they behave very much like the solutions in the single well potential.
Why does the energy level for small $$n$$ increase only a tiny amount between the near identical functions but a larger amount between the pairs? $$\psi_n(x)$$ has $$n$$ nodes, which is why the ground state cannot be degenerate: each node of the wave function corresponds to a place where $$\psi'_n(x)$$ must be nonzero and is hence associated with some kinetic energy. But since the first two eigenfunctions are almost identical the difference is slight, while the step towards the next pair corresponds to getting some node in the low-potential region with much bigger kinetic energy. Eventually the number of nodes become so many that the steps even out, and the near degeneracy disappears.