Solving a Parabolic Function Near the Minimum of the Morse Potential

[capslock]

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

 

[nrqed]

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

 

[kyle8921]

As a fellow scientist, I understand your frustration with trying to solve this problem. The Morse Potential is a commonly used function in physics and chemistry, and its behavior near the minimum is of great interest. To show that the potential energy is a parabolic function near the minimum, we can use mathematical techniques such as the Taylor series expansion.

First, let's review the definition of a parabolic function. A parabolic function is a type of quadratic function where the highest degree of the polynomial is 2. This means that the function can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants.

Now, let's take a closer look at the Morse potential equation, U = E_0 (1-exp(-a(r-r_0))^2. By expanding the exponential term using the Taylor series, we can rewrite the equation as U = E_0 (1- (1 + (-a(r-r_0)) + (-a(r-r_0))^2/2 + ...) ^2.

Simplifying this further, we get U = E_0 (1 - 2(-a(r-r_0)) + (-a(r-r_0))^2 + ...). Notice that as we approach the minimum of the potential, which is at r = r_0, the higher order terms become negligible. This is because as r becomes closer to r_0, the term (-a(r-r_0))^2 will become very small compared to the other terms.

Therefore, we can approximate the Morse potential near the minimum as U = E_0 (1 - 2(-a(r-r_0))). This is a parabolic function in the form of y = ax^2 + bx + c, where a = 2E_0a, b = -2E_0a, and c = E_0.

In conclusion, by using the Taylor series expansion, we have shown that near the minimum of the Morse potential, the potential energy is a parabolic function. This result is important in understanding the behavior of the potential energy near the minimum and its implications in various physical and chemical phenomena. I hope this explanation helps in your understanding of this problem. Keep up the good work, James!