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Homework Help: Hamiltonian Operator

  1. Oct 14, 2007 #1
    For an atom with one electron and nuclear charge of Z, the Hamiltonian is:

    [tex] H=-~\frac{\nabla^{2}}{2}~- ~\frac{Z}{r}~ [/tex]

    1) show that the wavefunction:

    [tex]\Psi_{1s}=Ne^{-Zr}[/tex]

    is an eigenfunction of the Hamiltonian

    2) find the corresponding energy

    3) find N, the normalisation constant

    In spherical polar coordinates:

    [tex]\nabla^{2}\Psi_{1s}=~\frac{1}{r^{2}}~(~\frac{d}{dr}~[r^{2}~\frac{d\Psi_{1s}}{dr}])[/tex]


    by applying H to the wavefunction, i think i've shown that it's an eigenfunction:

    1)

    [tex]H\Psi_{1s}=~\frac{-Z^{2}r^{2}}{2}~Ne^{-zr} - ~\frac{Z}{r}~[/tex]

    [tex]H\Psi=n\Psi [/tex]

    where n is the eigenvalue, and the bit on the end:

    [tex]- ~\frac{Z}{r}~[/tex]

    doesn't matter right, i've still shown it's an eigenvalue?

    2)

    to find the corresponding energy, don't I need to know N first?

    3)

    to find N, am I right in thinking:

    [tex] N^{2} \int \Psi* \Psi dx = 1[/tex]

    [tex]N^{2} \int e^{-2zr} dx = 1[/tex]

    [tex]~\frac{-N^{2}}{2z}~e^{-2zr}=1[/tex]

    [tex]N=\sqrt{~\frac{-2}{e^{-2zr}}~}[/tex]

    but i think i must have gone wrong somewhere, i mean that doesn't look right. Once i've found N, how do i find the corresponding energy do i just plug N into:


    [tex]H\Psi_{1s}=~\frac{-Z^{2}r^{2}}{2}~Ne^{-zr} - ~\frac{Z}{r}~[/tex]

    ie, [tex]H\Psi_{1s}[/tex] = the corresponding energy?

    i'd appreciate any help, thanks
     
  2. jcsd
  3. Jan 15, 2009 #2
    You haven't multiplied

    [tex]\frac{Z}{r}[/tex]

    right, it should be

    [tex]\frac{Z}{r}~Ne^{-Zr}[/tex]
     
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