How do i find minimum for this potential?

[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?

 

[siddharth]

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?

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

 

[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...

 

[siddharth]

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...

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}) ?

 

[wormhole]

siddhart , so sorry
i simply already did some rearrangemnet:

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

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

 

[Integral]

Wormhole,
To maximize the readability of your equations spend a few minutes reading https://www.physicsforums.com/showthread.php?t=8997"

 

[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 . 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}

 

[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

 

[siddharth]

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

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.

 

[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