Limitations on radial wavefunction for electron in an atom

  1. What are the limitations on the radial wavefunction for electron in an atom?

    For instance, of the following, which cannot be the radial wave function, and why?

    1.) [tex]e^{-r}[/tex]
    2.) [tex]\sin(br)[/tex]
    3.) [tex] \frac{1}{r}[/tex]

    Thanks!!
     
  2. jcsd
  3. dextercioby

    dextercioby 12,303
    Science Advisor
    Homework Helper

    For an eigenfunction of the hamiltonian to describe physically acceptable states,it needs some requirements:
    a)Continuous oner the entire domain (0,+oO)
    b)Annulation in asymptotic limit (condition for bound states)
    c)Lebesgue square integrabilty.

    The only function of your three to have them all is 1),but 2) describes scattering states for the electrons of the atom.It doesn't annulate in asymptotic limit (r->+oO) and is not Labesgue square integrable.But it is an eigenfunction of the Hamiltonian.Because it's not normalizable,it does not fulfill the requirement expressed in the first postulate,therefore it cannot describe quantum states with discrete spectrum for the Hamiltonian.These functions can be normalized in Dirac generalized sens,implying distributions.

    Daniel.
     
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