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Homework Help: Derive Planck Length in terms of c, G, and h_bar

  1. Feb 20, 2009 #1
    1. The problem statement, all variables and given/known data

    a. The first part of this problem was to derive the escape speed (Vesc) for a star of mass M and radius R.

    b. The second part is where I am having trouble. It says to equate the Vesc calculated above and derive a formula for Planck length in terms of c, G, and h_bar.

    2. Relevant equations

    This is from the Heisenberg uncertainty section for position and momentum so:

    [tex]\Delta[/tex]X * [tex]\Delta[/tex]P = h_bar/2

    E = mc2

    3. The attempt at a solution

    The escape velocity I calculated from part a was Vesc = [tex]\sqrt{2GM/R}[/tex]

    Equating Vesc to C gives:

    C = [tex]\sqrt{2GM/R}[/tex]

    Am I supposed to assume that [tex]\Delta[/tex]p = Mc and solve for [tex]\Delta[/tex]x using the Heisenberg uncertainty principle?

    I have tried a couple of different methods and I can't get this to simplify down to the form I need. I am pretty sure this is an easy problem, I just am not seeing the trick.
  2. jcsd
  3. Feb 20, 2009 #2


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    Homework Helper

    According to Wikipedia the Planck length is defined as sqrt(hG/c^3)
    I think it is just a way to get units of meters out of those fundamental constants.
    I don't think it has a physical meaning, except in quantum mechanics which doesn't seem to apply to the problem you have.
  4. Feb 20, 2009 #3
    Ok I think I am getting close to this one:


    M=[tex]\frac{c^2 R}{2G}[/tex]

    Using the Heisenberg Principal:
    [tex]\Delta[/tex]X [tex]\Delta[/tex]P = [tex]\frac{h_bar}{2}[/tex]

    [tex]\Delta[/tex]P = c M

    Solving this for [tex]\Delta[/tex]X
    [tex]\Delta[/tex]X = [tex]\frac{G h_bar}{c^3 R}[/tex]

    How do I get the square root and get rid of the R? :frown:
  5. Sep 21, 2010 #4
    Gravity level
    c =under root of (2GM/R)
    where R=L (Planck's Length) on gravity level
    so c =under root of (2GM/L)

    Uncertainty principle
    uncertainty in momentum x uncertainty in position = reduced Planck constant/2
    mv x L = reduced Planck constant/2
    here on quantum level, Uncertainty in position is Planck length and velocity is c (velocity of light)
    mcL= reduced h/2
    m=reduced h/(2cL)

    put the value of m in value of c in gravity level
    c square = 2 G reduced h/(2cL*R)
    c square = 2 G reduced h/(2cL*L)
    where R=L (Planck length on gravity level)
    Derive L (Planck length) from above equation.
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