Commit ccc133ef authored by jensj's avatar jensj

Update formula to what is used in the code.

parent 201008f6
......@@ -140,15 +140,14 @@ radial Schrödinger equation:
= \epsilon u.
We want to solve this equation on a non-equidistant radial grid with
`r_g=r(g)` for `g=0,1,...`. Inserting `u(r) = a(g) r^\ell`, we
`r_g=r(g)` for `g=0,1,...`. Inserting `u(r) = a(g) r^{\ell+1}`, we
get:
.. math::
\frac{d^2 a}{dg^2} (\frac{dg}{dr})^2 r^2 +
\frac{da}{dg}(r^2 \frac{d^2g}{dr^2} + 2 \ell r \frac{dg}{dr}) -
2 \ell a +
2 r^2 (\epsilon - v) a = 0.
\frac{da}{dg}(r^2 \frac{d^2g}{dr^2} + 2 (\ell+1) r \frac{dg}{dr}) -
2 r^2 (v - \epsilon) a = 0.
Including Scalar-relativistic corrections
......@@ -168,11 +167,19 @@ where the relativistic mass is:
M = 1 - \frac{1}{2c^2} (v - \epsilon).
With `u(r) = a(g) r^\ell` and `\kappa = (dv/dr)/(2Mc^2)`:
With `u(r) = a(g) r^\alpha`, `\kappa = (dv/dr)/(2Mc^2)` and
.. math::
\alpha = \sqrt{\ell^2 + \ell + 1 -(Z/c)^2},
we get:
.. math::
\frac{d^2 a}{dg^2} (\frac{dg}{dr})^2 r^2 +
\frac{da}{dg}(r^2 \kappa \frac{dg}{dr} + r^2 \frac{d^2g}{dr^2} +
2 \ell r \frac{dg}{dr}) +
[2 M r^2 (\epsilon - v) - 2 \ell + \kappa (\ell - 1) r] a = 0.
2 \alpha r \frac{dg}{dr}) +
[2 M r^2 (\epsilon - v) +
\alpha (\alpha - 1) - \ell (\ell + 1)
+ \kappa (\alpha - 1) r] a = 0.
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