Defining wavefunctions

How to define a trial wavefunction?

Single particle problem:

Hydrogen atom

Scrhodinger Equation (SE): \((\frac{\hbar^2}{2m} \nabla^2 - \frac{Zke^2}{r})\psi(\vec{r}) = E\psi(\vec{r})\) \(\psi(\vec{r}) = k(r)P_{me}(\theta_g \phi)\) \(x,y,z -> r, \theta, \phi | r \in [0, \infty), \theta \in[0, \pi], \phi \in [0, 2\pi]\)

Local energy: \(\hbar = 1, m = 1, ke^2 = 1\) (atomic units) \(\frac{1}{k(r)} (- \frac{d^2}{dr^2} - \frac{2}{r}\frac{d}{dr} -\frac{Z}{r})k(r)\) First two terms inside parenthesis is kinetic energy, Z/r is potential energy

What could go wrong:

  • The derivatives \(\frac{d^m}{dr}k(r)\) are finite for all r (else it would converge)
  • k(r) is finite for all r
  • lim r -> 0: Massive problem because of division by r.
    • Must ensure \((-\frac{Z}{r}\frac{d}{dr}r - \frac{Z/r}k) = 0\)
    • \(-\frac{Z}{k(r)r} \frac{d}{dr}r = \frac{Z}{r} \frac{k}{k}\)
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