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

Hydrogen probably homework

  • Thread starter syang9
  • Start date
61
0
1. Homework Statement
Griffiths 4.14: What is the most probably value of r, in the ground state of hydrogen? (Hint: First you must figure out the probability that the electron would be found between r and r+dr.

Here is the posted solution:

http://www.glue.umd.edu/~syang9/problem%204.14%20solution.PNG [Broken]

I don't understand the approach; why does multiplying the probability density by [tex] \[
4\pi r^2 dr\] [/tex] give the probability that the electron will be found between r and r+dr? Why do we differentiate p(r)? In general, I thought the way to find the probability of a particle being in a particular energy eigenstate was to take the inner product of the general wavefunction with the energy eigenstate..

[tex]
\[
\left| {c_n } \right|^2 = \int_{ - \infty }^\infty {\Psi ^* (\overrightarrow {\bf{r}},t) \cdot \psi (r){\rm{ }}dr}
\]
[/tex]

But the general wave function is a sum which involves [tex] c_n [/tex].. so I don't understand what governs which approach to take..
 
Last edited by a moderator:

Answers and Replies

61
0
I hope this question doesn't fall into the abyss of unanswered questions, forever forgotten by all..
 
Dick
Science Advisor
Homework Helper
26,258
618
It takes longer to fall into the abyss of unanswered questions than two hours. Be patient. The solution is doing exactly what the hint proposes. It's finding the probability density of the wave function as a function of r by integrating over the angular coordinates. That's where the 4*pi*r^2 comes from, it's what you get from that integration together with the volume element in spherical coordinates. Then you maximize it, hence the derivative.
 
Last edited:
61
0
So I want to figure out at what point the probability density is a maximum; that's why I set the derivative to zero. OK, I get it, thanks!
 
Last edited:
61
0
(Also, yes, you are right. I will try to be more patient.)
 

Related Threads for: Hydrogen probably homework

  • Last Post
Replies
6
Views
5K
  • Last Post
Replies
9
Views
1K
Replies
1
Views
7K
Replies
12
Views
3K
Replies
7
Views
6K
Replies
7
Views
864
Replies
3
Views
8K
Replies
1
Views
758
Top