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A double-well potential has two local minima separated by a potential energy barrier. A symmetric confining example isIts minima are at $x=\pm a$ and the central barrier has height $\lambda a^4$. Near either minimum, writing $x=\pm a+\xi$ gives $V\simeq 4\lambda a^2{\xi }^2$, a harmonic potential with local angular frequency $\sqrt{8\lambda a^2/m}$ for mass $m$.

For this symmetric one-dimensional bound-state problem, the lowest two energy eigenstates can be chosen as an even function and an odd function. They are delocalized across both wells, with splitting $\Delta E=E_1-E_0>0$ produced by quantum tunnelling. Define orthonormal combinations $|L⟩=(|0⟩+|1⟩)/\sqrt{2}$ and $|R⟩=(|0⟩-|1⟩)/\sqrt{2}$. For well-separated wells with a sufficiently high barrier, these approximate states localized on the left and right. An initial $|L⟩$ evolves within this doublet withThe first full transfer takes $\pi \hslash /\Delta E$. A two-mode approximation for a Bose gas uses analogous left/right orbitals; its validity also requires suppression of higher energy levels and appropriate control of interactions. Two local minima alone do not guarantee that every arbitrary nonconfining potential has two discrete bound levels.

Macroscopic quantum self-trapping is a nonlinear regime of a bosonic Josephson junction in which an initial population imbalance does not undergo the full sign-changing tunneling oscillations expected from a simple linear two-state model. Interactions alter the relation between relative phase and population flow.

An experiment with two weakly linked condensates in a double-well potential measured both the density distribution and the relative phase inferred from interference fringes. Their combination distinguished nonlinear self-trapping from merely observing unequal populations at one instant.^[ref]

The two-mode approximation for a Bose gas retains the real localized orbitals:It is valid when the lowest doublet is well separated from higher energy levels, and the temperature and interaction-induced changes of the orbitals are small on that separation. More explicitly, if $\Delta$ is the energy gap to discarded orbitals, sufficient low-energy conditions are $k_{B}T,t,N{U}_{0}I\ll \Delta$, where $I=\int d^{3}r|{\phi }_L{|}^4$. A deep double-well potential suppresses interwell interaction overlaps. Large $N$ alone does not justify the truncation: increasing $N$ at fixed coupling can violate the interaction condition.

Let $ϵ=(E_0+E_1)/2$ and use the even and odd eigenfunctions to evaluate the projected one-particle Hamiltonian operator:ThusThe same quantum tunnelling matrix element is $t=-\int d^{3}r{\phi }_L{h}_0{\phi }_R$. Reflection symmetry makes $\int |{\phi }_L{|}^4=\int |{\phi }_R{|}^4=I$. Neglecting mixed contact interaction integrals leavesThe last step is the normal-ordered boson pair count, following from the canonical commutation relation. Consequently the two-site Bose-Hubbard model has precisely the requested coefficientsandHere $\mu$ is the coefficient obtained by rewriting the fixed-particle number Hamiltonian operator, rather than an independently prescribed thermodynamic chemical potential. Adding a reservoir term $-{\mu }_{\mathrm{res}}\hat{N}$ would replace it by $\mu +{\mu }_{\mathrm{res}}$.

The negligible-overlap instruction applies to the mixed contact interaction integrals while the small quantum tunnelling matrix element is retained. Exactly disjoint smooth orbitals for a local $h_0$ would also give $t=0$; setting every overlap-related matrix element identically to zero would discard the physical splitting one is trying to describe.

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.