Solving a Parabolic Function Near the Minimum of the Morse Potential

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capslock
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I have the equation for the Morse potential, U = E_0 (1-exp(-a(r-r_0))^2. I'm asked to show that near the minimum of the curve the potential energy is a parabolic function. I've tried to play around with the taylor series with no hope! :( :(

Many thanks, James
 

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capslock said:
I have the equation for the Morse potential, U = E_0 (1-exp(-a(r-r_0))^2. I'm asked to show that near the minimum of the curve the potential energy is a parabolic function. I've tried to play around with the taylor series with no hope! :( :(

Many thanks, James

It is indeed just a simple Taylor expansion! Can you show your work?
The potential vanishes at r=r_0 and the derivative of the potential also vanishes at r=r_0. The second derivative does not vanish at that point so you get that U(r) is approximately U''(r=r_0)/2 r^2 so a parabolic function.

Pat