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Project a quantized field onto two low-energy orbitals. A sufficient regime has temperature, hopping and interaction shifts small compared with the energy gap to discarded orbitals. Small interwell contact interaction integrals can then be neglected while retaining a small quantum tunnelling matrix element. The projected field's operator commutator is the two-mode projection operator kernel, rather than a complete spatial Dirac delta function.

Two localized orbitals coupled by quantum tunnelling and on-site interactions give the smallest Bose-Hubbard model. A symmetric double-well potential with lowest one-particle energies $E_0,E_1$ has $t=(E_1-E_0)/2$ and common diagonal energy $(E_0+E_1)/2$. At fixed total particle number, that common diagonal term is constant. A fixed-number two-mode condensate gives a simple variational method for this model, while stronger interactions can require occupation-correlated trial quantum states.

For a finite jump in a one-dimensional potential energy, with constant mass and no delta potential, both the wavefunction and its first spatial derivative are continuous. Integrating the time-independent Schrödinger equation across a vanishing interval makes the jump in ${\psi }^{\prime }$ vanish, since the integral of $(V-E)\psi$ vanishes. A jump in $\psi$ itself would produce a derivative of a Dirac delta function in ${\psi }^{\prime \prime }$, with no matching term in the equation. The second derivative can jump. This matching allows transmission and quantum tunnelling; a finite step is not a hard wall.