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.(adsbygoogle = window.adsbygoogle || []).push({});

I started with the harmonic oscillator but have no reached the Gaussian one:

V = -V_{0}exp(-x^{2})

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 V_{0}. On the attached graph I have done this for V_{0}=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) = -V_{0}+ V_{0}x^{2}

Plugging this into my dimensionless Schrödinger equation I get:

½∂^{2}ψ/∂x^{2}+ (-V_{0}+ V_{0}x^{2})ψ = Eψ

I thought aha. The x^{2}-term can just be put in the harmonic oscillator form if we pick k=2V_{0}and the -V_{0}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)=½x^{2}for the harmonic oscillator. Now I have 2V_{0}in front of that. How will this constant effect my energies?

And is all this even the right procedure?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Gaussian potential

Loading...

Similar Threads for Gaussian potential |
---|

I Tunneling through a finite potential barrier |

A Vacuum in QFT: Fock space or effective potential? |

I Probabilities Associated with Sudden Changes in Potential |

I Lho potential |

B Uncertainty of a Gaussian wavepacket |

**Physics Forums | Science Articles, Homework Help, Discussion**