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Total energy of Coulomb model of Hydrogen atom

  1. Sep 5, 2013 #1
    1. The problem statement, all variables and given/known data
    Hi, my question is regardng a Coulomb model of an H atom specified with Hamiltonian operator, Hhat, by spherical coordinates of energy eigenfunction
    ψ2,1,-1 (r,θ, ∅) =(1/ 64∏a02)1/2 r/a0 e-r/2a0 sinθ e-iθ

    Principal quantum numer n = 2
    orbital an mom l = 1
    magnetic quantum number m = -1

    I must specify magnitude of orbital ang momentum, Lhat2, and z-component of orbital ang momentum as well as total energy.


    2. Relevant equations
    so;
    Lhat2 = l (l + 1) hbar2
    Lz = m hbar (1/ √2∏) eim∅



    3. The attempt at a solution
    Lhat2 = 1 (1 + 1)hbar2
    Lhat2 = 2 hbar2

    Lz = m hbar (1/ √2∏) eim∅
    Lz = -1 hbar / √2∏ ei-1∅
    = -1 hbar

    but what about total energy?
    I'm lost in the forest again and can't see the wood for the trees.
     
  2. jcsd
  3. Sep 5, 2013 #2

    mfb

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    Staff: Mentor

    There is a relation between principal quantum number and energy that you should know.
     
  4. Sep 6, 2013 #3
    Hmmm!

    Well, En = (n + 1/2) hbar ω0

    and

    En = n22 hbar2 / 2mL2
    where principal quantum number n = 0, 1, 2 . . .

    but I can't evaluate ω0 and I don't have the mass for the other one . . do I?
     
  5. Sep 6, 2013 #4
    Ah!
    n = principal quantum number
    m = mass of electron
    L2 = square of magnitude of orbital ang momentum.

    so . . . .

    E = 22 ∏2 hbar2 / 2(9.1x10-31) 2 hbar2
    E = -2∏hbar/m

    er! surely too messy and probably wrong.
     
  6. Sep 6, 2013 #5

    mfb

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    Why? Your formula (edit: wait, those are the wrong formulas) assumes an infinite central mass, which is usually a reasonable approximation in a hydrogen atom.

    The total energy of the system does not matter unless you want to consider special relativity.
     
    Last edited: Sep 6, 2013
  7. Sep 6, 2013 #6

    TSny

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    Does either of these formulas apply to the hydrogen atom?
     
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