1. Boundary value problems:
so far we have considered initial value problems
e.g.
some problems require the solution to be fixed at the
end-points
e.g.
Examples include
temperature distribution in a rod
Schrodinger equation
soap films

2. Shooting method: To solve with boundary conditions
assume an initial value for
solve as an initial value problem
vary until solution “hits” target

3. Time independent Schrodinger equation:
2nd order ode
solve for and E with boundary conditions
normalization
as the Schrodinger equation is linear in , and homogeneous we may fix the normalization after finding a solution. i.e.
if is a solution, so is
use the symmetry of V(x) to give another boundary condition

4. Symmetric potential:
if V(x)=V(-x) then the wavefunction has definite parity
even
odd
this fixes the boundary condition at the origin
even
odd
due to the linearity of the Schrodinger equation we may always
choose
Use these as initial conditions in a shooting algorithm
Only for certain values of E (the eigen values) will

5. Example: simple harmonic oscillator
introduce some rescaled variables
then we have that
solve as an initial value problem, using wavefunctions of
either parity, to find the allowed values of

6. The potential is symmetric so look for even/odd solutions.
shooting method:
even,

7. The potential is symmetric so look for even/odd solutions.
shooting method:
odd,

8. Matching methods
if the potential is not symmetric then the wavefunction does not have definite parity
but we know that
approximate these conditions by
For some far to the left, and far to the right. (Must check that answers are not sensitive to your choices.)
pick another value such that

9. now solve from both directions
find solution in the region , using initial conditions
find solution in the region , using final conditions
(i.e. using the shooting method from to the left, )

10. rescale one of the solutions such that
vary E until the solutions have the same slope at

11. then the eigenfunction is given by
the normalized wavefunction is then given by

12. Example: simple harmonic oscillator
pick an energy, En=4.1
pick co-ordinate range,
integrate from the left and right using shooting

13. Example: simple harmonic oscillator
rescale one of the solutions,
this is not smooth, it has a “V” kink, so choose another energy

14. Example: simple harmonic oscillator
try another energy, En=6.1
this is also not smooth, but has an “inverted V” kink.
choose an energy between that with the “V” kink and that with the “inverted V” kink.

15. Example: simple harmonic oscillator
After a bit of trial and error in choosing En we get the final eigenfunction.

16. Homing in on En:
In both the shooting method and the matching method we find that the required solution is in some region
For the shooting method we see that picking the wrong En means the solution diverges to either or This determines E1 and E2.
for the matching method we see that picking the wrong En means the solution has either a “V” kink, or an “inverted V” kink; this determines E1 and E2.
By taking another trial energy in this range are able to home in on the correct solution, up to some desired accuracy. A natural choice is half way between E1 and E2.