Approximaths

Quantum Mechanical Interpretation

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The differential equation presented in the previous posts

\displaystyle y'' + (1 - x)y = 0

appears in quantum mechanics when a particle is subjected to a linear potential. The time-independent Schrödinger equation is

\displaystyle -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi(x) = E\psi(x)

Consider a linear potential of the form

\displaystyle V(x) = V_0 - Fx

where F is a constant force. The Schrödinger equation then becomes

\displaystyle -\frac{\hbar^2}{2m}\psi''(x) + \bigl(V_0 - Fx\bigr)\psi(x) = E\psi(x)

Rearranging the terms gives

\displaystyle \psi''(x) + \frac{2m}{\hbar^2}\bigl(E - V_0 + Fx\bigr)\psi(x) = 0

We introduce the turning point x_0 (in quantum mechanics, it marks the transition between an oscillatory (classically allowed) region and an exponentially decaying (classically forbidden) region) defined by

\displaystyle E - V_0 + Fx_0 = 0
\displaystyle x_0 = \frac{V_0 - E}{F}

The equation then becomes

\displaystyle \psi''(x) + \frac{2mF}{\hbar^2}(x - x_0)\psi(x) = 0

Introducing the dimensionless variable

\displaystyle z = -\left(\frac{2mF}{\hbar^2}\right)^{1/3}(x - x_0)

we finally obtain

\displaystyle \frac{d^2\psi}{dz^2} - z\psi = 0

Thus, the equation

\displaystyle y'' + (1 - x)y = 0

is, after a suitable rescaling and translation of the independent variable, an Airy equation. It describes, in particular, the behavior of a quantum wavefunction near a turning point, i.e. a position where

\displaystyle E = V(x)

In a region where E > V(x), the solution is oscillatory, whereas in a region where E < V(x), it exhibits exponential behavior. The Airy function provides the transition between these two regimes (see fig. below).

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