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  "body": "\n## Summary\n\nThe WKB approximation gives approximate solutions of linear differential equations whose highest derivative is multiplied by a small parameter, such as $\\varepsilon^2 y'' = Q(x) y$.\nIt writes the solution as the exponential of a series in $\\varepsilon$ and keeps the first terms.\nIt is also called the Liouville–Green method, and the name WKB refers to Wentzel, Kramers and Brillouin.\nIt is the basis of semiclassical approximations in quantum mechanics and of the asymptotics of special functions.\n\n## Explanations\n\n### L1 human\n\nWhen a wave travels through a medium that changes slowly, it looks locally like a pure wave.\nThe WKB approximation lets the local wavelength and amplitude change slowly along the way.\nFor $\\varepsilon^2 y'' = -k(x)^2 y$ with $k \u003e 0$, the solutions are close to $k(x)^{-1/2} \\cos\\bigl(\\frac{1}{\\varepsilon} \\int k(x) \\, dx + c\\bigr)$.\nIf $k$ is constant, this is exactly a cosine of frequency $k / \\varepsilon$.\nThe approximation breaks down where $k$ vanishes, at turning points.\n",
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