Analytical approach to 1D bound state problems
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Using a recently developed new analytic approach to solution of the 1D Schr¨odinger equation [Eleuch et al., Europhys. Lett. 89, 50004 (2010)], we derive a quantization rule determining the positions of bound state energy levels in an arbitrary potential. The obtained quantization formula is applied to a simple model problem and is found to be superior to the well known Bohr-Sommerfeld (JWKB) quantization rule. Contrary to the JWKB prescription, our new formula contains extra contributions arising from the classically forbidden regions. The corresponding analytic wavefunctions are smooth even at the classical turning points.