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Modified Bohr Atom Radii

  1. Apr 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Instead of electrons in the hydrogen atom experiencing the Coulomb potential, lets say they experience a potential in the form of [tex]V(r) = V_0(\frac{r}{R})^k[/tex], where [tex]V_0 > 0,\: R > 0, \:k > 0[/tex].

    With this information find an equation for the radii of the Bohr orbit with quantum number n.


    2. Relevant equations
    [tex]V(r) = V_0(\frac{r}{R})^k[/tex]

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

    [tex]F_{Coulomb} = F_{centripetal}[/tex]


    3. The attempt at a solution
    Under normal circumstances the equation for radii would be easy enough to derive from the fact that [tex]F_{Coulomb} = F_{centripetal}[/tex] for the electrons. So, I attempted to find the derivative of the electron's potential since it should be equal to the negative of Coulomb force. So,
    [tex]F_{Coulomb} = -\frac{dV(r)}{dr} = -\frac{k V_0 (\frac{r}{R})^{(k-1)}}{R}[/tex].
    Thus,
    [tex]-\frac{k V_0 (\frac{r}{R})^{(k-1)}}{R} = \frac{mv^2}{r}[/tex].

    And since angular momentum [tex]L = mvr = n\hbar[/tex],
    [tex]\frac{n^2\hbar^2}{mr^2} = -\frac{2 k V_0 (\frac{r}{R})^{(k-1)}}{R}[/tex]

    But at this point I don't see how I can get r on one side of the equation, so perhaps I approached the problem incorrectly or made a mistake somewhere. Any help will be greatly appreciated.


    EDIT: Okay, so my algebra seemed to be off and going back through I think I've come up with a viable answer.

    Given as I said before,
    [tex]\frac{k V_0 (\frac{r}{R})^{(k-1)}}{R} = \frac{mv^2}{r}[/tex].

    [tex]\frac{k V_0 (\frac{r}{R})^{(k-1)}}{R} = \frac{n^2 \hbar^2}{m r^3}[/tex]

    [tex]\frac{r^3 k V_0 (\frac{r}{R})^{(k-1)}}{R} = \frac{n^2 \hbar^2}{m}[/tex]

    [tex]\frac{r^{(2 + k)} k V_0}{R^k} = \frac{n^2 \hbar^2}{m}[/tex]

    [tex]r^{2 + k} = \frac{n^2 \hbar^2 R^k}{m k V_0}[/tex]

    [tex]r_n = \left(\frac{n^2 \hbar^2 R^k}{m k V_0}\right)^{(\frac{1}{2 + k})}[/tex]

    So, now I think this is the right answer. If anyone thinks differently, please correct me.
     
    Last edited: Apr 12, 2009
  2. jcsd
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