# How do i find minimum for this potential?

1. Dec 12, 2005

### wormhole

how do i find minimum for this potential:

u(r)=L^2/2*m*r^2 - b*exp(-ar)/r ?

b,a - constants
L - angular momentum
m - mass

u(r)=L^2/m+b*exp(-ar)*r(a*r-1)

(u(r*):=0 => r*=r*(a,b,L,m) )

i plotted the function for some values of b,a,L and m and i see that for certain values there is a minimum but i just can't figure out how do i find the right r.

can anyone help/give some hint?

2. Dec 12, 2005

### siddharth

If
$$U(r) = \frac{L^2}{2mr^2} - \frac{b e^{-ar}}{r}$$
then, what you have written for U'(r) is not correct.

3. Dec 12, 2005

### wormhole

ok, the correct expression for u(r):

u(r)=L^2/m-b*r*(a*r+1)exp(-a*r)

still i don't quite know how to proceed...

4. Dec 12, 2005

### siddharth

How is that the correct expression?
What is
$$\frac{d}{dr} (\frac{1}{r^2})$$?

and do you know how to find
$$\frac{d}{dr} (\frac{e^{-ar}}{r}) ?$$

Last edited: Dec 12, 2005
5. Dec 12, 2005

### wormhole

siddhart , so sorry
i simply already did some rearrangemnet:

$$\frac{du}{dr}=\frac{L^2}{mr^3} + \frac{b(are^{-ar} + e^{-ar})}{r^2}$$

the expression i wrote is when i do
u`(r):=0
and i multiply by r^3
so sorry for confusion

Last edited: Dec 12, 2005
6. Dec 12, 2005

### Integral

Staff Emeritus
Wormhole,

7. Dec 12, 2005

### siddharth

Ok, by solving U'(r)=0, you can find the critical points and then you can find the points of minima with the second derivative test.

On second thought, the equation you have to solve to find r looks really tough :surprised . I can find no obvious way to solve it.

For values of (ar)<<1, you could approximate by using the series expansion of $$e^{ar}$$

Last edited: Dec 12, 2005
8. Dec 12, 2005

### wormhole

i'll try to do that...
from the plots i did the r where u(r) is at minimum is very close to zero
i have to take only first term in $e^{-ar}$ expansion series otherwise i get an third order equation

Last edited: Dec 12, 2005
9. Dec 12, 2005

### siddharth

Can you post your solutions for r when you get it? This is an interesting problem and I want to check that I got the right answers so I can show my classmates.

Last edited: Dec 12, 2005
10. Dec 16, 2005

### wormhole

i asked my classmates about the solution and some say that there is no need to use expansion series...
i'm not sure what they did is right..so when i get the official solution(next week) i will give you a link to it