Okay, I have solved the schrödinger equation numerically by making it dimensionless (though Im still confused about this proces). And then approximating it on a finite interval and solving the resulting eigenvalue equation. This allows me to solve for the wave function of different potentials. I started with the harmonic oscillator but have no reached the Gaussian one: V = -V0exp(-x2) In one simulation I am asked to find the difference in energy between the ground state and the first excited state as a function of V0. On the attached graph I have done this for V0=1..2..3...10 Does it look right? I am then asked the following: Solve the problem analytically by taylorexpanding the potential. So I taylor expand around x=0 to second order and find: V(x) = -V0 + V0x2 Plugging this into my dimensionless Schrödinger equation I get: ½∂2ψ/∂x2 + (-V0 + V0x2)ψ = Eψ I thought aha. The x2-term can just be put in the harmonic oscillator form if we pick k=2V0 and the -V0 term will just shift the energy of the oscillator, not alter the difference between E1 and E0. But in thinking it over again there are some problems. With my dimensionless equation I just had V(x)=½x2 for the harmonic oscillator. Now I have 2V0 in front of that. How will this constant effect my energies? And is all this even the right procedure?