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The notion that the region form –infinity to infinity could be considered by a change of variables is explored in cofv.htm.

An unexpected result of taking the region from –infinity to infinity with a capped potential is the fact that the energy ground state is not zero at r=0. Then when I attempted to do the assignment at the end of this document, I found it impossible to converge on excited states. This led to the development of a variance minimization technique for solving the one-dimensional equation Schroedinger equation. Variance minimization files describe this technique. An annotated listing of the files leading up to this is in Welcome.htm.

A consideration of the method described below in curve fit notation so that a matrix is used to find the next solution is explored in the Neqns.htm, while the precise nature of the iterative solution below is considered in ItSol.htm.

When I attempted to do the assignment at the end of this document, I found it impossible to converge on excited states.

The one-d Schroedinger equation

in Rydbergs is

First consider the small r limit and assume that . Then

for . Next consider the large r limit

Thus the starting assumption is

The Fourier transforms of u and U are defined by

Fourier sums are considered in DFT.htm. The simplest Fourier sum starts at zero in r. This works but introduces a discontinuity in at least the first derivative at the origin making the sum more slowly convergent -- requiring a larger N -- than it needs with a better choice of range DFT- one_sided.

For r=-R/N, the first negative point this yields

As r®¥, u(r) and all of its derivatives go to zero, but this is not the case as r ® 0. For this reason u has been extended into the negative r region as so that

the integral is from -R/2 to R/2 in finding U(f). The function of r in equation 6 is

So that

Assignment

Note – Bob Coldwell was unable to make the following converge on excited states. The method that works is variance minimization.

Crude effective potential

Equation 1

This implies where a, b, c, and d can be determined to make the states have desired properties.

Figure 1 Zeff(r) ZEFF.FORwith a = b = c = d = 0.

The resulting effective potential is

Solve the 1-d Schroedinger equation for the 3s and 3p states with the above effective potential using the method outlined below.

Step 1 Initialization

Assume u is given by equation . Use a coarse grid e.g. 16 points. Calculate
u₁(r_i) on this grid. Use the FFT to find U₁(f_k) on this grid.

Step 2 Find C(f,u)

Calculate

As

Note that for real u

Step 3 solve for U in frequency space

Note that

Later on note that

Note that for real u(r)

Step 4 normalization

Find A from the condition

Step 5 find a new estimate of u

Note that this relation is merely a proof that u(r) is real, it makes no sense to truncate the fft.

Note that this needs to be on a symmetric range.

Step 6 test for convergence

Test for convergence, use Aitkin’s extrapolation very third step. If not converged go to step 2.

Step 7 double the number of points

If the number of points is >= N_max then stop. Otherwise use the FFT as described in DFT- Symmetric_range_in_r_and_f_with_FFT to double the number of points and then go to step 2