Double-well splitting and localized states

For a real symmetric potential on the full line with ordinary decaying bound-state conditions, bound eigenvalues are nondegenerate and the eigenfunctions have definite parity. The ground state has no interior nodes and can be taken positive. In a symmetric double well, its two lobes therefore have the same sign. The next state's opposite-sign lobes give odd parity and a central node. These facts depend on the connected one-dimensional problem; an infinite barrier splitting the domain changes the assumptions.

If the lowest pair is well separated in energy from higher states, construct orthonormal localized basis states $|L\rangle$ and $|R\rangle$ within that doublet. Their effective Hamiltonian has equal diagonal energy $E_0$ and an off-diagonal coupling chosen as $-J$ with $J>0$. This sign convention makes the symmetric state lower. The matrix is exact as a specified two-level model; treating it as an approximation to a particular spatial well requires verifying isolation of the doublet and computing its coupling.

An initial left state contains equal symmetric and antisymmetric amplitudes. Each amplitude acquires its own stationary phase under time evolution. Factoring out their common phase leaves a cosine left amplitude and an imaginary sine right amplitude. Squaring gives oscillating populations whose sum remains one. A stationary parity eigenstate itself never transfers its probability from left to right; time dependence requires the superposition. The apparent paradox disappears once an eigenstate is distinguished from a localized initial preparation.

The first complete transfer occurs when the relative phase becomes $\pi$. A full return needs another equal interval. Increasing barrier separation or height usually reduces low-energy wavefunction overlap and hence the splitting in a weak-overlap regime, making this time long. That qualitative connection does not make the effective coupling universal. A detailed spatial potential, or an independently justified approximation to it, is needed before a barrier height can be turned into a numerical transfer time.

If one well is offset by detuning $\delta$, diagonal energies become $E_0\pm\delta/2$. The gap is then $\sqrt{\delta^2+4J^2}$, but the largest possible transfer probability from a localized initial state drops to $4J^2/(\delta^2+4J^2)$. A larger gap therefore need not mean more complete tunnelling. Part of that gap can come from bias rather than coupling. At zero coupling the states remain localized even if their energies differ substantially.

An environment can further destroy the coherence between the doublet components. A statistical half-left, half-right mixture and a coherent equal superposition can share basis populations but differ in interference and later evolution. Additional excited states can create several beating frequencies after an energetic preparation. The clean sine-squared result is consequently a useful controlled model for an isolated near-resonant pair, not a claim that every double-well experiment performs perfect periodic transfer indefinitely.