Airy functions

A.7 Airy functions

Airy functions are the solutions of the diﬀerential equation:

\frac{d^{2} f}{d z^{2}} - z f = 0

There are two solutions, $Ai (z)$ and $Bi (z)$ ; the ﬁrst tends to zero as $z \to \infty$ , while the second blows up. Both are oscillatory for $z < 0$ .

The Mathematica functions for obtaining them are AiryAi[z] and AiryBi[z].

The asymptotic forms of the Airy functions are:

\begin{array}{rcl} A i (z) \overset{z \to \infty}{\to} \frac{e^{- \frac{2}{3} z^{3 ∕ 2}}}{2 \sqrt{π} z^{1 ∕ 4}} & and & A i (z) \overset{z \to - \infty}{\to} \frac{cos (\frac{2}{3} {|z|}^{3 ∕ 2} - \frac{π}{4})}{\sqrt{π} {|z|}^{1 ∕ 4}} \\ B i (z) \overset{z \to \infty}{\to} \frac{e^{\frac{2}{3} z^{3 ∕ 2}}}{\sqrt{π} z^{1 ∕ 4}} & and & B i (z) \overset{z \to - \infty}{\to} \frac{cos (\frac{2}{3} {|z|}^{3 ∕ 2} + \frac{π}{4})}{\sqrt{π} {|z|}^{1 ∕ 4}} \end{array}

The Schrödinger equation for a linear potential $V (x) = β x$ in one dimension can be cast in the following form

- \frac{ℏ^{2}}{2 m} \frac{d^{2} ψ}{d x^{2}} + β x ψ - E ψ = 0

Deﬁning $z = x ∕ x_{0}$ , with $x_{0} = {(ℏ^{2} ∕ (2 m β))}^{1 ∕ 3}$ , and $E = {(ℏ^{2} β^{2} ∕ (2 m))}^{1 ∕ 3} μ$ , and with $y (z) \equiv ψ (x)$ , this can be written

\frac{d^{2} y}{d z^{2}} - z y + μ y = 0

(see section A.11 for more on scaling.) The solution is

y (z) = C Ai (z - μ) + D Bi (z - μ) or ψ (x) = C Ai ((β x - E) ∕ (β x_{0})) + D Bi ((β x - E) ∕ (β x_{0}))

where $D = 0$ if the solution has to extend to $x = \infty$ . The point $z = μ$ , $x = E ∕ β$ is the point at which $E = V$ and the solution changes from oscillatory to decaying / growing.

The equation for a potential with a negative slope is given by substituting $z \to - z$ in the deﬁning equation. Hence the general solution is $ψ (x) = C Ai (- x ∕ x_{0} - μ) + D Bi (- x ∕ x_{0} - μ)$ , with $D = 0$ if the solution has to extend to $x = - \infty$ .

The ﬁrst few zeros of the Airy functions are given in Wolfram MathWorld.

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