• Support PF! Buy your school textbooks, materials and every day products Here!

The time average potential of neutral hydrogen atom

  • #1

Homework Statement


[/B]
The time-averaged potential of a neutral hydrogen atom is given by

latex.png


where q is the magnitude of the electronic charge, and
latex.png
being the Bohr radius. Find the distribution of charge( both continuous and discrete) that will give this potential and interpret your result physically.

Homework Equations


[/B]
latex.png


The Attempt at a Solution


[/B]
latex.png


latex.png
since
latex.png
and
latex.png


from product rule

latex.png


latex.png


now I'm stuck here no idea how to handle first term.
 
Last edited by a moderator:

Answers and Replies

  • #2
Ok I think I figure it out.

lets take derivative one term at a time using product rule

[itex]\frac{1}{r^{2}}\frac{\partial}{\partial r}\left[r^{2}e^{-\alpha r}\frac{\partial}{\partial r}\left(\frac{1}{r}\right)\right]=\frac{1}{r^{2}}\left[2re^{-\alpha r}\frac{\partial}{\partial r}\left(\frac{1}{r}\right)+r^{2}\left(-\alpha\right)e^{-\alpha r}\frac{\partial}{\partial r}\left(\frac{1}{r}\right)+r^{2}e^{-\alpha r}\frac{\partial^{2}}{\partial r^{2}}\left(\frac{1}{r}\right)\right][/itex]

putting [itex]\frac{\partial}{\partial r}\left(\frac{1}{r}\right)=-\frac{1}{r^{2}}[/itex]

[itex]\frac{1}{r^{2}}\left[2re^{-\alpha r}\left(-\frac{1}{r^{2}}\right)+r^{2}\left(-\alpha\right)e^{-\alpha r}\left(-\frac{1}{r^{2}}\right)+r^{2}e^{-\alpha r}\frac{\partial^{2}}{\partial r^{2}}\left(\frac{1}{r}\right)\right]=\frac{1}{r^{2}}\left[-2e^{-\alpha r}\frac{1}{r}+\alpha e^{-\alpha r}+r^{2}e^{-\alpha r}\frac{\partial^{2}}{\partial r^{2}}\left(\frac{1}{r}\right)\right][/itex]

[itex]=-2e^{-\alpha r}\frac{1}{r^{3}}+\alpha e^{-\alpha r}\frac{1}{r^{2}}+e^{-\alpha r}\frac{\partial^{2}}{\partial r^{2}}\left(\frac{1}{r}\right)[/itex]
 
Last edited:
  • #3
DEvens
Education Advisor
Gold Member
1,203
451
Is it exp(alpha r) or exp(- alpha r) ?

I have not checked your algebra. But you stopped with there still being a derivative. You are not done.
 

Related Threads on The time average potential of neutral hydrogen atom

  • Last Post
Replies
1
Views
648
  • Last Post
Replies
18
Views
2K
Replies
1
Views
1K
  • Last Post
Replies
18
Views
4K
Replies
1
Views
6K
  • Last Post
Replies
16
Views
5K
Replies
4
Views
6K
Replies
1
Views
1K
  • Last Post
Replies
1
Views
949
  • Last Post
Replies
4
Views
3K
Top