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I Why the energy inversely propotional to n^2 in Bohr model

  1. Sep 13, 2016 #1

    KFC

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    After reading some materials on Bohr model, I understand the model is more or less incorrect, especially in terms of "orbital". I just wonder if the energy expression is also wrong or not.

    In my text for general quantum theory, the energy about two neighboring level is given as ##\Delta E = E_n-E_{n-1} = \hbar\omega##
    which is ##n## independent. But in Bohr model about hydrogen, the quantized energy is given as
    ##E_n = -E_0/n^2##
    which is inversely proportional to ##n^2##. This will give, for example from level ##n## to ##n-1##

    ##
    E_n-E_{n-1} = \frac{-E_0}{(n-1)^2-n^2}
    ##
    which is ##n## dependent. I am quite confusing why is it. Is Bohr model about energy is correct for hydrogen or hydrogen-like atom?
     
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  3. Sep 13, 2016 #2

    Simon Bridge

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    The formula ##E_n = -E_0/n^2## is correct, and is due to the shape of the hydrogen potential.
    The hydrogen potential has a top, so the energies levels get closer together as they approach E=0 from below. For E > 0, the energy levels are continuous and the hydrogen atom is said to be "ionized".

    Your quantum theory text should not give ##\hbar\omega## increments in energy for the hydrogen potential - I suspect you have misread it - that spacing is for an approximation for other types of potential called "harmonic oscillator".

    Note: your equation for the transition energy is incorrect. Try again.

    See:
    http://astro.unl.edu/naap/hydrogen/transitions.html
     
  4. Sep 13, 2016 #3

    KFC

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    Thanks a lot. I think I misunderstood some context in the text. I always think the ##\Delta E=\hbar\omega## is universal for all quantum system. Your reply help me to recall the harmonic oscillator. So like what you said, the explicit form of energy for a quantum system is really depending on the potential, is that what you mean?

    Thanks anyway.
     
  5. Sep 13, 2016 #4

    Simon Bridge

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    That is correct - it is exactly the same for classical physics: the dynamics depends on the specific form of the potential. For instance, a ball rolling around the inside of a bowl - requires more kinetic energy to reach the same distance from the center in a steep sided bowl as for a shallow bowl.

    You can see this has to be the case if you consider the case of a repulsive potential, or the free-space potential (V=0 everywhere): does it make sense for the allowed energies in these situations to have the same separation as for bound-states?
     
  6. Sep 13, 2016 #5

    KFC

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    Thanks for clarifying it. I appreciate your help.
     
  7. Sep 13, 2016 #6

    Simon Bridge

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    No worries.
     
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