Tunneling splits a doublet without creating a decay rate

Assume two identical wells have one low-lying localized state each, with common approximate energy E_w. If their overlap is weak and higher local excitations are well separated, project the full Hamiltonian onto those two states, orthonormalizing them as needed. Choose their phases so the effective off-diagonal element is negative with magnitude J. The symmetric combination then has lower energy E_w−J, while the antisymmetric one has E_w+J. The doublet splitting is twice the coupling, not the coupling itself.

An initially localized left state is an equal superposition of these two energy eigenstates. Their phases evolve at slightly different rates, causing coherent transfer between the wells. The right-well probability is a sine squared whose argument contains Jt/ħ, and the first complete transfer occurs after πħ/(2J). This closed two-state model predicts recurring oscillations. Calling that inverse time a universal tunneling decay rate would discard the coherence and the recurrence without specifying a bath, an absorbing region, or another irreversible mechanism.

In a smooth barrier with two relevant turning points, the decaying WKB tail carries an amplitude suppression exp(−S/ħ), where S is the integral of the positive forbidden momentum between the points at the localized-state energy. The overlap coupling and doublet splitting inherit an exponential of this kind. A transmission probability through a similar barrier instead squares an amplitude and has the leading exponential exp(−2S/ħ). Confusing the two can change an estimate by exponentially many orders of magnitude.

The prefactor multiplying the splitting has energy units and contains normalization, well curvature, and matching information. It is not determined by the exponent alone. Low-lying ground doublets require particularly careful matching near quadratic well minima, and textbook prefactors can fail even when the leading logarithm is correct. Near the barrier top or when the wells are strongly coupled, the assumption of an isolated localized doublet and a large forbidden action can both break down.

In a path integral continued to imaginary time, the dominant tunneling path connects the two minima through a Euclidean classical solution. If the potential minimum is set to zero, its zero-Euclidean-energy action is the integral of √(2mV) from one minimum to the other. This path is not a real-time trajectory of a particle climbing an energetically forbidden hill. Its action organizes a saddle-point approximation, while fluctuations around it, including a translation zero mode, determine additional factors in the splitting.

The zero-energy instanton action and the WKB action evaluated at the localized-state energy use different endpoints and energies. Their exponential and prefactor organization must be matched consistently before comparing precise formulas. The quantum lab can illustrate a coherent doublet state: Reset and set both controls to θ=90° and φ=0°, predicting Bloch x=1; Reset and set θ=90° and φ=180°, predicting x=−1. If the basis labels the parity eigenstates, these opposite x directions represent localized combinations. Playback adds a fixed phase rate; it does not calculate a barrier action, choose a physical splitting, or produce irreversible decay.