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

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

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

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

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

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

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

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

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, )

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

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

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

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

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.

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

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.