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Accuracy of the Wentzel-Kramers-Brillouin (WKB) Quantization Condition for Symmetric Power Law Potentials

Frank Kampas
Organization: Physicist at Large Consulting
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It is well known that the WKB quantization condition gives the correct energy levels for the quantum mechanical simple harmonic oscillator (quadratic potential) and that it gives a good approximation in general for states with a large quantum number. For symmetric power law potentials V(x) \[Proportional] | x ^p | , it is shown that the WKB energy levels \[Proportional] (n + 1/2)^((2p)/(2+p)) where n is the quantum number. Furthermore, the value of the WKB prediction divided by the actual value for the ground state energy decreases with increasing p, with values of approximately 1.09, 1.0, 0.90 ,0.82, 0.75,and 0.25 for p = 1, 2, 3, 4, 5, \[Infinity]. The accuracy of the inverse WKB quantization condition for some of these potentials is also discussed.

*Applied Mathematics > Numerical Methods > Approximation Theory
*Mathematics > Calculus and Analysis > Differential Equations
*Science > Physics > Quantum Physics
*Wolfram Technology > Kernel > Numerics
*Wolfram Technology > Programming > Symbolic Computation

Quantum Mechanics, WKB Approximation, Quantized Energy Levels
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WKB_accuracy.nb (176.5 KB) - Mathematica Notebook