Airy functions for a linear potential

The Schrödinger equation for a linear potential has Airy functions as its solution. One of its interesting properties is that a translation of the origin can be compensated by a shift in energy level, so that in principle the solutions for any one energy suffice for all. The equation is

$$\frac{d^2\Psi}{dx^2} (\alpha x - E)\Psi,$$ = (327)

whose matrix coefficient, with respect to a pair of first order equations, would be

$$\left[ \begin{array}{cc} 0 & \alpha x - E \\ 1 & 0 \end{array} \right].$$ (328)

This matrix has a negative pair of eigenvalues, one of which is $\lambda = \surd(\alpha x - E)$; real or imaginary according to the relative sizes of $\alpha x$ and E. There are also eigenvector matrices, P for columns and P^-1for rows:

$$\begin{array}{ccc} P & = & \frac{1}{\surd2} \left[ \begin{array}{cc} \lambda & -\lambda \\ 1 & 1 \end{array} \right], \end{array} \begin{array}{ccc} P^{-1} & = & \frac{1}{\surd2} \left[ \begin{array}{cc} 1/\lambda & 1 \\ -1/\lambda & 1 \end{array} \right]. \end{array}$$ (329)

With this preparation, the WKB factorization embodied in equation (266) can be applied. We need the integrated eigenvalue to produces an angle

$$\phi \int_0^x\surd \overline{\alpha \sigma - E}\ d\sigma$$ = (330)

$$\frac{2\surd\alpha}{3} (\alpha x - E)^{\frac{3}{2}},$$ = (331)

whose rate of increase, a $\frac{3}{2}$ power, lies between linear and quadratic.

At first appearance, the solution of the Airy equation would be

$$\frac{1}{\lambda^2} \left[ \begin{array}{cc} \cos(\phi) & -\sin(\phi) \\ \sin(\phi) & \cos(\phi) \end{array} \right],$$ Z(x) = (332)

which combines an inverse first power drop in amplitude with the square-root like increase in frequence with increasing distance. That is an asymptotic behavior near infinity; near the classical turning point, there is an important correction to the diagonal matrix P^-1MP, namely

$$P^{-1}\frac{dP}{dx} \frac{\alpha}{4\lambda^2} \left[ \begin{array}{cc} 1 & -1 \\ -1 & 1 \end{array} \right],$$ = (333)

which requires its own treatment. As a scalar multiple of a constant matrix, the solution is an exponential of that matrix, this time with the multiplier

$$\theta \int_0^x \frac{\alpha d\sigma}{4(\alpha \sigma - E)}$$ = (334)

$$\frac{1}{4}\ln(\alpha x - E).$$ = (335)

  

Figure 19: wave functions for a linear potential, which is solved with Airy functions. Generally, they act like $\sin(r^{3/2})/r$ and $\cos(r^{3/2})/r$.

Figure 19 shows the result of a numerical integration of the Airy equation, choosing the two basic solutions as the one similar to a sine (initial value 0, derivative 1) together with the one similar to a cosine (initial value 1, derivative 0). Of course, calculating the exponential of a logarithm gives the argument back, which is a way to avoid the singularity implied by the logarithm. It would be more in keeping with the spirit of complex variable theory to give the energy a small imaginary component, replacing E by $E+i\varepsilon$, and narrowly avoiding the singularity. The main preoccupation is to get the trigonometrric functions on one side of the singularity to connect smoothly with the hyperbolic functions on the other side.

Splitting the coefficent matrix for the Schrödinger equation has mainly historical interest, given the ease of performing numerical integrations nowadays. Nevertheless, it is often useful to have an approximation in terms of more ordinary functions, either for symbolic calculations or as a starting point for mixed numerical computations.

  

Figure 20: wave functions for a linear potential in the phase plane.

Figure 20 shows the same result in the phase plane, which is the natural domain of the Prüfer transformation.