Particle in 1-D Potential

1. Sep 22, 2010

dats13

1. The problem statement, all variables and given/known data

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

2. Relevant equations

The Time Independent Schrodinger Equation (TISE):

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

3. 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.

2. Sep 22, 2010

Thaakisfox

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

3. Sep 22, 2010

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 $$x^{2}$$.

4. Sep 23, 2010

Thaakisfox

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

5. Sep 23, 2010

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.