Morse oscillator ($E_n=A(n+1/2)-B(n+1/2{)}^2$)

The quantum Hamiltonian with kinetic energy $p^2/(2\mu )$ and Morse potential has bound energies of the displayed form, where $A=\hslash {\omega }_e$, $B=A^2/(4D_e)>0$ and ${\omega }_e=a\sqrt{2D_e/\mu }$. Adjacent energy differences are $A-2B(n+1)$, distinct and positive over the physical bound-level range. A finite truncation with nonzero adjacent dipole couplings therefore satisfies connected nondegenerate transition chain generates special unitary control. Controllability of such a truncation does not by itself prove controllability on the full space including dissociation states.

A plot of successive vibrational spacings against a vibrational index. Extrapolation toward zero spacing estimates a dissociation limit under an assumed spacing law. The integration bounds and whether the energy is measured from the potential minimum or vibrational ground state must be specified.