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Bohr model

  1. Oct 10, 2004 #1
    Here's my problem:
    Show that the speed of an electron in the nth Bohr orbit of hydrogen is (alpha*c)/n, where alpha is the fine structure constant. What would be the speed in a hydrogenlike atom with a nuclear charge of Ze?

    We didn't talk about the fine structure constant in class, so could someone explain to me what it is? Hints on how to show that speed = alpha c/n would also be appreciated.

  2. jcsd
  3. Oct 10, 2004 #2

    Kane O'Donnell

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    Science Advisor


    In the Bohr model we assume that angular momentum is quantised:

    [tex] L = mvr = n\hbar [/tex]

    From this you can find the expression for the tangential velocity of the electron. You then need to find the expression for the Bohr radius for a particular value of n, which turns out to be (for Z = 1, for Hydrogen-like atoms just replace e^2 with Z(e^2)):

    [tex] r_{n} = \frac{4\pi\epsilon_{0}\hbar^{2}n^2}{me^2} [/tex]

    When you sub in for r you get:

    [tex] v_{n} = \frac{e^2}{4\pi\epsilon_{0}\hbar} [/tex]

    From this you should be able to work out what the fine structure constant is - just compare the equation you were given to the one above. In undergrad physics courses the name "fine structure constant" is often applied to a few dimensionless constants that all look similar. It's just a number that happens to arise in a lot of Quantum Mechanical situations. You'll see it a fair bit :)


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