QM · Ch 8 · WKB bound states of V(x)=α|x|

The Bohr–Sommerfeld condition quantizes closed orbits by requiring the action integral $\oint p \, dx$ around a complete cycle to equal integer multiples of $h$. For this V-shaped potential, the particle oscillates symmetrically between two classical turning points, so you'll integrate the momentum $p(x) = \sqrt{2m(E - \alpha|x|)}$ across one full bounce and set it equal to $(n + \tfrac{1}{2})h$ (the half-integer accounts for the phase at turning points). This turns a transcendental integral into an algebraic equation for $E_n$.

The Bohr–Sommerfeld condition quantizes closed orbits by requiring the action integral $\oint p \, dx$ around a complete cycle to equal integer multiples of $h$. For this V-shaped potential, the particle oscillates symmetrically between two classical turning points, so you'll integrate the momentum $p(x) = \sqrt{2m(E - \alpha|x|)}$ across one full bounce and set it equal to $(n + \tfrac{1}{2})h$ (the half-integer accounts for the phase at turning points). This turns a transcendental integral into an algebraic equation for $E_n$.