Finding Excited States of a Particle in a 1-D Potential

[dats13]

Homework Statement

The quaestion asks to determine the ground an first excited state of the wavefuntion of a particle in a 1-D potential given by $$V(x)=B\left | x \right |$$.

Homework Equations

The Time Independent Schrödinger Equation (TISE):

$$-\frac{\hbar}{2m}\frac{d^{2}\Psi }{dx^{2}}+V\Psi=E\Psi$$

The Attempt at a Solution

I substituted the potential into the TISE and with some rearraging of terms I get the following differential equation.

$${\Psi}''+\frac{2m}{\hbar^{2}}(E-B\left | x \right |)\Psi=0$$

This is where I'm stuck. I don't know how to solve this equation because of the potential is dependent on $$x$$. Any suggestions would be greatly appreciated.

 

[Thaakisfox]

The general solutions will be Airy functions (or Bessel functions of order 1/3).

 

[dats13]

Ok, but I don't see how I would get bessel's equation. I'm guessing I would need to multiply the equation first by $$x^{2}$$.

 

[Thaakisfox]

Its much easier to get Airy's equation. Check how you can get that ;)

 

[dats13]

It turns out for this question I don't actually need to solve for the wavefunction. I just need to determine the curvature of it. Than from there I can get a rough sketch of the excited states, which is all the question asks for. I should have posted the full question, although I know what I need to do now. Thanks for your input Thaakisfox.