Wave function on slope potential

[jostpuur]

What are solutions to

$$\psi''(x) = (a_0 + a_1 x)\psi(x)$$
?

First idea I've had was that I could try some kind of perturbation with respect to the $a_1$ variable, so that

$$\psi(x) = A_1e^{\sqrt{a_0}x} + A_2e^{-\sqrt{a_0}x} + \psi_1(x)$$

would be an attempt. But I couldn't find anything useful for $\psi_1$ when $a_1\neq 0$, so it got stuck there.

Second idea was to use Fourier transform to transform the problem into this form

$$\phi'(x) = (b_0 + b_1x^2)\phi(x)$$

Unfortunately the solutions

$$\phi(x) = Be^{b_0x + \frac{1}{3}b_1x^3}$$

don't have converging Fourier transforms, so that doesn't help either.

 

[phyzguy]

The solution is called an Airy function, and it is a much-studied solution to Schrödinger's equation in the semi-classical approximation. In addition to the link below, you might try searching on "semiclassical approximation" or "WKB method".

http://en.wikipedia.org/wiki/Airy_Function

 

[jostpuur]

I see and appreciate.