Expectation value <p> of the ground state of hydrogen

  • #1
Warda Anis
3
0

Homework Statement


How should I calculate the expectation value of momentum of an electron in the ground state in hydrogen atom.

Homework Equations


Untitled.png

The Attempt at a Solution


I am trying to apply the p operator i.e. ##-ihd/dx## over ##\psi##. and integrating it from 0 to infinity. The answer I am getting is ##ih/{2(pi)a^3}##
 

Attachments

  • Untitled.png
    Untitled.png
    4.3 KB · Views: 1,515

Answers and Replies

  • #3
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
11,995
5,020
You are actually trying to calculate the expectation value of px. To do that, you must express the ground state wavefunction explicitly as a function of ##x## so that you can take the derivative. You have to do a 3d integral.
 
  • #4
Warda Anis
3
0
So I did the following to integrate it in 3d:$$ \iiint_V \Psi^* (-i\hbar) \frac {d\Psi} {dr} r^2 sin\theta dr d\theta d\phi$$

The final answer i am getting is ##\frac {i\hbar}{a_b}## which does not look right because it is imaginary and momentum operator is hermitean. I can't figure out what mistake I am doing
 
  • #5
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
11,995
5,020
You are assuming that in spherical coordinates ##p_r=-i \hbar \frac{\partial}{\partial r}.## It is not quite that. Do some research on the internet to find out what it is and why.
 

Suggested for: Expectation value <p> of the ground state of hydrogen

  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
12
Views
1K
Replies
5
Views
1K
Replies
11
Views
2K
Replies
1
Views
1K
Replies
3
Views
23K
Replies
12
Views
2K
Top