A bound system cannot separate into a specified set of free constituents without acquiring additional energy. Its stability also depends on the available decay routes and interactions. Attraction alone is not a complete account of a persistent material object.
An energy landscape
Imagine a relative separation r between two constituents. A model potential U(r) summarizes how their interaction energy changes with separation after other degrees of freedom have been treated appropriately. A minimum allows small disturbances to produce a restoring force. An escape threshold tells us how much energy is needed to reach widely separated configurations.
These are different pieces of information. A parabola has a minimum but rises without limit, so its ideal model has no dissociation threshold. A realistic approximation to a molecular bond must eventually allow separation. The Morse form is a useful example with a finite asymptote.
A well with an exit
Set q = a(r − r₀), where a is an inverse length and r₀ is the equilibrium separation. Choose ar₀ = 2 so that the displayed separations are positive. With depth D > 0, take
U(q) = D[(1 − e⁻ᑫ)² − 1].
The minimum is −D at q = 0, and the potential approaches zero as separation grows. For classical relative motion with positive reduced mass, kinetic energy is nonnegative. An allowed position must therefore satisfy U(q) ≤ E. For −D < E < 0, there are two turning points and the separation remains bounded.
Explore the binding threshold
How far can the constituents separate? Change the well depth and total relative energy. The calculation is classical; it does not display quantized molecular levels.
Solid green: potential U/D₀. Dashed purple: total energy E/D₀. The coloured segment marks allowed separations on the energy line. An arrow means the allowed interval continues beyond the plot. Both axes keep fixed scales.
Read the turning points
Writing e = E/D, solve (1 − e⁻ᑫ)² = 1 + e. The inner turning point is −ln(1 + √(1 + e)). A finite outer turning point, −ln(1 − √(1 + e)), exists when −1 < e < 0. At e = −1 they coincide at the equilibrium point.
At the threshold e = 0, the outer point moves to infinity. An outward trajectory can keep separating with speed tending to zero; infinity is not reached at a finite time. Above threshold, the trajectory still has an inner turning point but no finite upper limit on separation. The model contains no friction or additional collision that could change E.
Increasing D while holding E/D fixed preserves the dimensionless turning positions, but increases the absolute energy scale. Near equilibrium, U ≈ −D + Da²(r − r₀)², so the small-oscillation angular frequency is a√(2D/μ), with reduced mass μ. Shape, binding energy and response time are connected but distinct.
Stability requires a physical model
A real molecule has quantum states, rotational motion and possible couplings to radiation or other particles. Its vibrational ground state is not the classical point at the bottom of the well. A potential minimum by itself also does not rule out every alternative reaction or decay channel.
For a specified quantum problem, binding concerns energies below the appropriate separation threshold, and stability requires examining accessible transitions. A metastable state may persist because escape is slow even when another state has lower energy. A deeper theory must calculate the interaction and the available channels, not merely draw an attractive curve.
Binding in a generative account
The book's treatment of composite mass requires internal motion and interaction energy to enter the collective description. Its statistics and mass-gap discussion also distinguishes a finite spectral separation from a physical excitation gap that survives the relevant limits.
For SMM/DTT, the next constructive step is to obtain a calibrated interaction from specified generative operations and use it to predict several properties of the same bound system. The Morse calculation supplies a transparent comparison model. Its depth and length scale are inputs, not constants derived from inner time.
Sources and further reading
- P. M. Morse, Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels, Physical Review 34, 57–64 (1929). The present demonstration uses the potential for an explicit classical calculation.
- Mohamed Haj Yousef, Composite mass and internal motion and Statistics and the mass gap.
- Continue with How composite systems become wholes.