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airy10

# airy10 - Physics 216 Non-Relativistic Quantum Mechanics II...

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Unformatted text preview: Physics 216 Non-Relativistic Quantum Mechanics II Spring 2010 Airy functions and their properties Airy functions are deﬁned as solutions to the following diﬀerential equation: d2 ψ − zψ (z ) = 0 . dz 2 The two independent solutions are the Airy functions of the ﬁrst and second kind: ψ (z ) = C1 Ai(z ) + C2 Bi(z ) , where C1 and C2 are constant coeﬃcients to be determined by the boundary conditions of the problem. The Airy functions are related to Bessel functions of one-third order: Ai(z ) = Bi(z ) = z 1/2 2z 3/2 I−1/3 3 3 z 3 1/2 − I1/3 2z 3/2 3 2z 3/2 3 , , I−1/3 2z 3/2 3 + I1/3 where Iν (z ) is the Bessel function of imaginary argument. The most important properties of the Airy functions for our purposes are the following asymptotic expansions, valid in the limit of real z → ∞: 1 2 Ai(z ) ∼ √ z −1/4 exp − z 3/2 2π 3 1 π 2 Ai(−z ) ∼ √ z −1/4 cos z 3/2 − π 3 4 1 2 Bi(z ) ∼ √ z −1/4 exp + z 3/2 π 3 π −1 2 Bi(−z ) ∼ √ z −1/4 sin z 3/2 − π 3 4 REFERENCE: N.N. Lebedev, Special Functions and Their Applications (Dover Publications Inc., Mineola, NY, 1972). ...
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