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Simplification in Schrodinger derivation

  1. Jan 29, 2013 #1
    This is not a homework question. This is not for a course. However, I got a warning for posting such questions elsewhere, so, I suppose I must post them here.

    1. The problem statement, all variables and given/known data
    The following is an excerpt of the derivation of the Schrodinger equation. After deriving the Klein-Gordon equation, the relativistic total energy is approximated to arrive at the Schrodinger equation.

    2. Relevant equations

    [itex]E = mc^2\sqrt{1+\frac{p^2}{m^2c^2}}[/itex]
    [itex]\approx mc^2\left(1+\frac{1}{2}\frac{p^2}{m^2c^2}\right)[/itex]


    3. The attempt at a solution

    Well, frankly, I do not see how they went from the first step to the second step. Where did the 1/2 come from? How does the removal of the square root effectively approximate this? I am not seeing it. Was a conjugate used and multiplied somehow?
     
  2. jcsd
  3. Jan 29, 2013 #2

    TSny

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  4. Jan 29, 2013 #3

    Mute

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    It may also help you to play around with some of these plots to get a feel for how well the approximations work. For example, letting ##x = p/(mc)##, take a look at a comparison of the two curves, plotted with wolframalpha:

    http://www.wolframalpha.com/input/?i=Plot[{Sqrt[1+x^2],1+x^2/2},{x,0,1}]

    Play around with some other examples of the binomial approximation as well.
     
  5. Jan 29, 2013 #4
    Thank you! This is perfect. I do not know how I managed to get this far without having heard of the binomial approximation, and having tutored math for years. I guess I am one of the lucky ten thousand.
     
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