Classical-limit formula for matrix elements between bound states of distinct one-dimensional potentials

In quantum mechanics, evaluating transition matrix elements between bound states of two distinct one-dimensional potentials is a common challenge. This problem frequently arises in molecular physics as Franck-Condon factors between different electronic states, as well as in the study of motional states of atoms trapped in one-dimensional optical lattices. When potentials support a high density of bound states, finding relative transition strengths becomes difficult. This work addresses the problem generically by deriving a classical-limit formula (CLF) for matrix elements between bound states of distinct one-dimensional potentials, bypassing the need for explicit wave function overlap integrals in the appropriate limit.

The study considers two slowly varying one-dimensional potentials, $U_1(x)$ and $U_2(x)$, each supporting a high density of bound states characterized by large node numbers ($n_i \gg 1$). The matrix element $M$ involves an integration over a slowly varying operator function $f(x)$ multiplied by the two real, normalized wave functions. Using the Wentzel-Kramers-Brillouin (WKB) approximation for the wave functions, the integral is evaluated primarily in the vicinity of a special point called the common oscillation point (COP), where the local wavenumbers of both states are equal. Through a second-order expansion of the phase difference around the COP and subsequent averaging over continuous energy variables in the classical limit, the authors arrive at the final closed-form CLF expression for $|M|^r$.

To test the validity of the derived formula, the authors first check a specific overlap case where $f(x) = 1$ and $r = 2$. By summing the squared matrix elements over all states of one potential, the result correctly evaluates to unity, confirming the internal consistency of the CLF. Furthermore, the CLF is tested using specific functional forms of the potentials based on the second Pöschl-Teller differential equation, which afford exact analytical solutions for bound-state energies and wave functions. By taking the classical limit of large parameter ratios, the numerically computed matrix elements exhibit a clear and strong convergence toward the predictions of the analytical CLF.