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wkb - P443 WKB I D.Rubin Connection Formulae The WKB...

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P443 WKB I D.Rubin February 18, 2008 Connection Formulae The WKB approximation falls apart near a turning point. Then E - V 0 so 1 p → ∞ . And because the momentum goes to zero the wavelength gets very long and the approximation is only valid if the wavelength is short compared to the distance over which the potential changes. But if want to determine bound state energies, we need to be able to match wave functions at the turning points. The strategy to overcome this limitation of the WKB wave functions at the turning points is to 1. Linearize the potential at the turning point ( x = 0). V ( x ) = V (0)+ x dV dx 2. Solve Schrodinger’s equation exactly near the turning point for the linear potential There will be two cases to consider. One where dV dx > 0 and the particle has positive kinetic energy to the left of the turning point ( x < 0), and negative kinetic energy to the right ( x > 0). The other case is when dV dx < 0 and then there is negative kinetic energy to the left and positive to the right. We consider first dV dx > 0. - ¯ h 2 2 m d 2 ψ dx 2 + ( V (0) + xV 0 ) ψ = d 2 ψ dx 2 = - 2 m ¯ h 2 ( E - V (0) - xV 0 ) ψ = - p 2 ¯ h 2 ψ The turning point is at x = 0 and that is where V ( x ) = E . So V (0) = E and we have d 2 ψ dx 2 = α 3 where α 3 = 2 m ¯ h 2 V 0 . Then if we define the dimensionless parameter z = αx we have d 2 ψ dz 2 = which is Airy’s equation and the general solution is ψ p = aAi ( x ) + bBi ( x ) 1
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ψ p is referred to as the patching wave function since its sole purpose is to patch together the WKB wave functions on each side of the turning point. 3. Determine the WKB wave function in the region of the linear potential. It will be different on each side of the turning point. To the left of the turning point for case one, the WKB wave function is ψ l ( x ) = A p e ( x ) + B p e - ( x ) and φ ( x ) = 1 ¯ h Z x 0 p ( x 0 ) dx 0 = Z x 0 α 3 2 ( - x ) 1 2 dx 0 = - 2 3 ( - αx ) 3 2 (1) where p ( x ) = q 2 m ( E - V ) q 2 m ( - xV 0 ) = ¯ 3 2 ( - x ) 1 2 (Since we are to the left of the turning point x < 0 and the argument of the square root is positive.) The WKB wave function to the left of
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