Potential well

A region of lower potential energy surrounded by higher potential energy; a particle must gain sufficient energy to escape classically.

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.

A confining Quantum Hamiltonian on $x>0$ with $a>0$, potential $a\sqrt{x}$ and a Dirichlet boundary condition at zero has natural length $\ell =({\hslash }^2/(2ma){)}^{2/5}$ and energy $E_{\ast }=({\hslash }^2{a}^4/(2m){)}^{1/5}$. Rescaling $x=\ell y$ gives $H/E_{\ast }=-d^2/d{y}^2+\sqrt{y}$, so every energy eigenvalue scales as $E_{\ast }$. The potential derivative diverges at the wall even though the potential itself is finite there.