LinearPotential - Linear potential February 8, 2008 Linear...

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Linear potential February 8, 2008 Linear Potential - Gravity The Schrodinger equation for the wave function of a bouncing ball is - ¯ h 2 2 m d 2 ψ dx 2 + mgxψ = (1) where we assume a perfectly elastic collision of the ball with the floor. So V ( x ) = for x < 0, and V ( x ) = mgx for V > 0. There is zero probability to find the ball at x < 0 so ψ (0) = 0. If we define a characteristic length, l 0 = ¯ h 2 2 m 2 g ! 1 3 and energy E 0 = ¯ h 2 mg 2 2 ! 1 3 then x = yl 0 and E = ±E 0 where y and ± are dimensionless and substitution into Equation 1 gives - d 2 ψ dy 2 + = ±ψ (2) Finally with the substitution of z = y - ± , Equation 2 becomes - d 2 ψ dz 2 + = 0 (3) Equation 3 is Airy’s equation and the solutions to it are Airy functions. (Some of the properties of Airy functions are summarized in Chapter 8 of Griffiths. For more details see http://dlmf.nist.gov/contents/AI/index.html) The general solution to Airy’s equation is ψ ( z ) = aA i ( z ) + bB
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LinearPotential - Linear potential February 8, 2008 Linear...

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