Some Examples

Here are some screen captures of various behaviors for the harmonic oscillator, having a symmetric potential, and the Morse oscillator, having an asymmetric potential.

Here is a screen capture of the desired asymptotic behavior for an harmonic oscillator. The harmonic oscillator is an example of a symmetric potential and thus has either Even or Odd wave functions.The energy is 0.5. This is the ground state and may be recognized as such by the lack of nodal points on the x axis. The wave function actually gives divergent behavior (not shown) for larger positive and negative values of x.

Here is solution for the harmonic oscillator potential. This is an example of a symmetric potential and thus has either Even or Odd wave functions.The solution shown has an energy of 0.501 and is higher than the acceptable value of 0.5. The solution is not acceptable as a wave function. Note the divergent behavior having crossings at both positive and negative x. This indicates too positive an energy and it should be reduced.

harmonic oscillator, energy too negative

Here is another solution for the harmonic oscillator potential. This is an example of a symmetric potential and thus has either Even or Odd wave functions.The Even solution shown has an energy of 0.499 and is lower than the acceptable value of 0.5. The solution is not acceptable as a wave function. Note the divergent behavior witout crossing the x axis in the regions where asymptotic behavior is desired. This indicates an energy which is too negative and it should be made more positive.

Here is a screen capture of a solution for the Morse potential (asymmetric). It is unacceptable, as a wave function, because it lacks asymptotic behavior. Because you have to obtain asymptotic behavior for both positive and negative x you will have too choose values for both the initial slope and the energy. The slope should be adjusted at this point and not the energy. Note that the x axis is crossed for the positive x axis where asymptotic behavior is desired, indicating too positive an energy, and not crossed at negative x, indicating too negative an energy. Adjust the slope until similar behavior is obtained for both positive and negative x asymptotic regions.

Here is a screen capture of a solution for the Morse potential (asymmetric). It is unacceptable, as a wave function, because it lacks asymptotic behavior. Because you have to obtain asymptotic behavior for both positive and negative x you will have too choose values for both the initial slope and the energy. The energy should be adjusted at this point and not the slope. Note that the x axis is crossed for both positive and negative x axes (in the regions where asymptotic behavior is sought) indicating too high an energy. When adjusting the energy the function at positive x will be affected differently than at negative x leading to a need to further refine the slope.