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Checking my reasoning on deriving fine sructure constant alpha

  1. Mar 26, 2014 #1
    1. The problem statement, all variables and given/known data

    From Bohr’s model of hydrogen atom, derive the ratio of the electron velocity in the ground state of hydrogen atom to the speed of light, which is the fine structure constant α, in terms of fundamental physical constants such as h, m, and c.


    2. Relevant equations

    Kinetic energy in a Bohr atom: [itex]\frac{Ze^2}{r^2}[/itex]
    kinetic energy of a body revolving around a center: [itex]\frac{mv^2}{r}[/itex]
    momentum of a body revolving around a center: [itex]L=mvr[/itex]


    3. The attempt at a solution

    So we set the kinetic energy equal to that of the Bohr atom (based on the Coulomb force)

    [itex]\frac{mv^2}{r}=\frac{Ze^2}{r^2}[/itex]

    Since we are dealing with hyrogen Z=1, so [itex]\frac{mv^2}{r}=\frac{e^2}{r^2}[/itex] and [itex]r=\frac{e^2}{mv^2}[/itex]

    Bohr's model says the angular momentum around a hydrogen atom is [itex]\frac{nh}{2\pi}=n\hbar[/itex]

    The total energy of the atom is [itex]T+U = E_T[/itex]

    and angular momentum can be expressed: [itex]mvr = \frac{nh}{2\pi}[/itex]
    so [itex]v = \frac{nh}{2\pi m r}[/itex] which we can substitute in to the equation for r and get

    [itex]r = \frac{e^2}{m(\frac{n h}{2\pi mr})^2}=\frac{4\pi^2 e^2 m r^2}{n^2 h^2}[/itex] which yields [itex]r = \frac{n^2 h^2}{4\pi^2 e^2 m}[/itex]

    That's the Bohr radius. We can plug that into the original relation we had for kinetic energy.

    [itex]\frac{mv^2}{r}=\frac{Ze^2}{r^2}[/itex]

    which gets us

    [itex]\frac{n^2 h^2}{4\pi^2 e^2 m r^2}=\frac{e^2}{mv^2}[/itex]

    and moving it all around algebraically

    [itex]v^2 = \frac{e^2 4\pi^2 e^2 m r^2}{n^2 h^2 m}[/itex]

    [itex]v = \frac{e^2 2\pi r}{n h}[/itex] and we divide that by c to get α.

    So my question is if I did this correctly. (I sense that someone will tell me I did something wrong, but I want to see if its conceptual or arithmetical mistake)
     
  2. jcsd
  3. Mar 27, 2014 #2

    DrClaude

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    Staff: Mentor

    It won't be a fundamental physical constant if it depends on ##r## and ##n##.
     
  4. Mar 27, 2014 #3
    OK, but since it is hydrogen (n=1) if I plug the expression for r in to the one I got for v:

    [tex]v=\frac{e^2 2 \pi}{n h} \frac{n^2 h^2}{4 \pi^2 e^2 m} = \frac{n h}{2 \pi m}[/tex]

    so that [tex]\alpha = \frac{n h}{2 \pi m c }[/tex] and since n=1

    [tex]\alpha = \frac{h}{2 \pi m c } = \frac {\hbar}{mc} [/tex]

    is that correct? (I figure n had to be 1 since we are talking about hydrogen and a ground state)
     
  5. Mar 27, 2014 #4

    DrClaude

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    Staff: Mentor

    No. You made a mistake somewhere. Start from
    and substitute in the value of ##r## you got.
     
  6. Mar 27, 2014 #5
    OK, doing that,

    [itex]v=\frac{nh}{2\pi m r}[/itex]

    and

    [itex]r=\frac{n^2 h^2}{4\pi^2 e^2 m}[/itex]

    So putting them together

    [tex]v=\frac{nh}{2 \pi m } \frac{4 \pi^2 e^2 m}{n^2 h^2} = \frac{2 \pi e^2}{n h}[/tex]

    which would mean that

    [tex]\frac{v}{c} = \frac{2 \pi e^2}{n h c} = \frac{e^2}{\hbar c} = \alpha[/tex]

    yes? assuming i got it righ this time I didn't need to do the second set of steps in my original derivation, I could have gone from when I got v originally rather than needing to go all the way to r.

    (also, it's ok to assume n=1, right? We are talking about H).
     
  7. Mar 27, 2014 #6

    DrClaude

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    Staff: Mentor

    Correct, although you should get rid of the ##n## before getting to this point (or at least mention why it disappeared here).

    Indeed, I don't know why you did that! Your reasoning should go like this: You find an equation for ##v##, which has a dependence on ##r##. You then try to resolve this, and fortunately you have another equality which involves ##r##. You solve that one, substitute ##r## in your equation for ##v##, and see what you get. In this case, the job is then done!
     
  8. Mar 27, 2014 #7
    Yes, I think I got too hung up on needing the Bohr radius itself, which isn't necessary for this particular problem.
     
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