Straight-forward quantum question

[quasar_4]

Homework Statement

A particle of mass m is confined by a 1-dimensional potential of the form V(x) = -V0*exp(-x^2/a^2). Assume that V0 is large enough that there are at least two bound states. Sketch the wavefunction of the ground state and the first excited state, clearly indicating the parity and asymptotic behavior of each. You are not asked here to solve the Schroedinger equation.

Homework Equations

The Attempt at a Solution

Ok... so the potential is a Gaussian.

Since the potential is not infinite, we need the total energy E to be less than V to have a bound state (right)?

My questions is: is the first excited state just the derivative of V? I guess it's not clear to me how to deal with this without solving for solutions, unless there's a simple relationship between the potential and the first excited state. The only way I can really think of doing this is to solve H*psi = E*psi, apply boundary conditions, and see what solutions I have. But I'm supposed to do this without solving that equation...

 

[chrispb]

What do the solutions to, say, the simple harmonic oscillator look like as you increase the energy? Why do they look this way? Are there any patterns they follow that might be true in general?