Simplification in Schrodinger derivation

[TheFerruccio]

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.

Homework Statement

The following is an excerpt of the derivation of the Schrödinger equation. After deriving the Klein-Gordon equation, the relativistic total energy is approximated to arrive at the Schrödinger equation.

Homework Equations

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

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?

 

[TSny]

See Binomial Approximation

 

[Mute]

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.

 

[TheFerruccio]

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.