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. |
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. |