Commit ccc133ef by 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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