Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.12

Particle in a Gravitational Field

Physics — Landau & Lifshitz · particle-in-gravitational-field

Problem

The Airy function Ai(z)\text{Ai}(z) satisfies d2Aidz2=z Ai(z)\frac{d^2 \text{Ai}}{dz^2} = z\, \text{Ai}(z) (a) Show that for large positive zz, the Airy function has the asymptotic form Ai(z)∼12π z1/4exp⁡(−23z3/2)\text{Ai}(z) \sim \frac{1}{2\sqrt{\pi}\, z^{1/4}} \exp\left(-\frac{2}{3}z^{3/2}\right) (b) Show that for large negative zz (writing z=−∣z∣z = -|z|), the Airy function oscillates as Ai(−∣z∣)∼1π ∣z∣1/4sin⁡(23∣z∣3/2+π4)\text{Ai}(-|z|) \sim \frac{1}{\sqrt{\pi}\, |z|^{1/4}} \sin\left(\frac{2}{3}|z|^{3/2} + \frac{\pi}{4}\right) (c) Use the result of (b) to derive the WKB quantization condition for the gravitational well.

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