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The semiclassical Wentzel–Kramers–Brillouin (WKB) method applied to one-dimensional problems with bound states often reduces to the Sommerfeld–Wilson quantization conditions, the cyclic phase-space integrals ... . It turns out that this formula gives the exact bound-state energies for the Morse oscillator with ... . The requisite integral can be reduced to ... , in which ... and ... are the classical turning points ... . The integral can be done "by hand", using the transformation ... followed by a contour integration in the complex plane, but Mathematica can evaluate the integral explicitly, needing only the additional fact that ... The result reads ... , which can be solved for ... to give ... , ... , in units with ... . The highest bound state is given by ... , where ... represents the floor, which for positive numbers is simply the integer part. The values of ... , ... , and ... used in this Demonstration are for illustrative purposes only and are not necessarily representative of any actual diatomic molecule.
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