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Expression for Bohr Radius using Quantum Mechanics; In Commutator Relations Chapter

  1. Jan 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Determine an expression for the Bohr radius (a[itex]_{0}[/itex] from the following approximation. The electron moves to the nucleus to lower its potential energy,

    V(r) = -[itex]\frac{e^{2}}{r}[/itex]

    If the electron is in domain 0[itex]\leq[/itex]r[itex]\leq[/itex][itex]\bar{r}[/itex], then we may write Δp≈[itex]\frac{\hbar}{\bar{r}}[/itex], with corresponding kinetic energy [itex]\frac{\hbar^{2}}{2m\bar{r}^{2}}[/itex]. With this information estimate a[itex]_{0}[/itex] (Bohr radius) by minimizing the total energy.


    2. Relevant equations

    Uncertainty ΔpΔx≈[itex]\frac{\hbar}{2}[/itex]
    [itex]\bar{r}[/itex] is the Mean of r -------> (I think)
    [itex]\hat{H}[/itex][itex]\varphi_{n}[/itex]=E[itex]_{n}[/itex][itex]\varphi_{n}[/itex] or with x-hat and x
    For an electron moving with momentum p 2[itex]\pi[/itex]rp=nh
    E = K.E + V(x)
    [[itex]\hat{A}[/itex], [itex]\hat{B}[/itex]]=[itex]\hat{A}[/itex][itex]\hat{B}[/itex]-[itex]\hat{B}[/itex][itex]\hat{A}[/itex]
    [[itex]\hat{A}[/itex], [itex]\hat{B}[/itex]]=0 if they they commute
    [[itex]\hat{x}[/itex], [itex]\hat{p}[/itex]]=i[itex]\hbar[/itex]
    And I have no idea where this equation came from but I found it in the book and used it for my solution:
    [itex]\frac{e^{2}}{r}[/itex]=[itex]\frac{p^{2}_{\vartheta}}{mr^{2}}[/itex]=[itex]\frac{n^{2}\hbar^{2}}{mr^{2}}[/itex] where e is the electric charge of the electron and p[itex]_{\vartheta}[/itex]=mr[itex]^{2}[/itex][itex]\dot{\vartheta}[/itex]







    3. The attempt at a solution


    E = [itex]\frac{p^{2}_{\vartheta}}{2m\bar{r}^{2}}[/itex]-[itex]\frac{e^{2}}{\bar{r}}[/itex] .... i
    From the "Centripetal Condition" (The equation I got from the book) (r)[itex]\frac{e^{2}}{r^{2}}[/itex]=(r)[itex]\frac{p^{2}_{\vartheta}}{mr^{3}}[/itex]=[itex]\frac{p^{2}_{\vartheta}}{m\bar{r^{2}}}[/itex] .... ii

    sub ii into i

    E = -[itex]\frac{p^{2}_{\vartheta}}{2m\bar{r}^{2}}[/itex] ..... iii

    Now from Centripetal Condition:
    r=[itex]\frac{n^{2}\hbar^{2}}{me^{2}}[/itex] where p[itex]_{\vartheta}[/itex]=n[itex]\hbar[/itex] ..... iv

    Put iv into iii

    E = -[itex]\frac{p^{2}_{\vartheta}}{2m}[/itex]([itex]\frac{me^{2}}{n^{2}\hbar^{2}}[/itex])[itex]^{2}[/itex]=-[itex]\frac{me^{4}}{2n^{2}\hbar^{2}}[/itex] and let ℝ=[itex]\frac{m_{e}e^{4}}{2\hbar^{2}}[/itex]=13.6eV
    Then E[itex]_{n}[/itex]=[itex]\frac{ℝ}{n^{2}}[/itex]

    So for n=1, which is the lowest possible state of potential energy, E[itex]_{1}[/itex]=13.6eV

    and for n=1 from equation iv

    r[itex]_{1}[/itex]=a[itex]_{0}[/itex]=[itex]\frac{n^{2}\hbar^{2}}{me^{2}}[/itex]=5.24x10[itex]^{-9}[/itex]cm=Bohr's radius

    The problem is this was too straightforward and simple. I did not correctly use the mean value of [itex]\bar{r}[/itex], because it is a function and I am to find an expression for the Bohr radius.
    I used very little of the information provided.
    Also I just assumed the lowest energy state, whereas the question says that the electron just goes to a lower energy state.
    Also this question is in the chapter where they define the commutator relationships and compatible operators, and really the whole chapter is about that. I find it fishy the solution I provided doesn't use any of that chapters material.


    I need a new approach to this problem and am out of ideas. Please help.
     
    Last edited: Jan 28, 2012
  2. jcsd
  3. Jan 29, 2012 #2

    vela

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    Re: Expression for Bohr Radius using Quantum Mechanics; In Commutator Relations Chapt

    The idea here is that the electron is confined to a region of size ##\bar{r}##, so applying the uncertainty principle gives you an estimate of the magnitude of its momentum, namely ##p \approx \hbar/\bar{r}##, and therefore of its kinetic energy. The total energy of the electron is thus given as a function of ##\bar{r}## by
    $$E(\bar{r}) = \frac{\hbar^2}{2m\bar{r}^2} - \frac{e^2}{\bar{r}}.$$ The problem is simply asking you to find the value of ##\bar{r}## for which the total energy of the electron is minimized.
     
  4. Jan 29, 2012 #3
    Re: Expression for Bohr Radius using Quantum Mechanics; In Commutator Relations Chapt

    I see it now, thank you!
     
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