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

Density of states in QM

  • Thread starter eoghan
  • Start date
  • #1
200
1
Hi. I'm studying the transition rates between a state a and a state b in the continuos level.
In the book "Physics of atoms and molecules" by Bransden and Joachain it is said:
We have to calculate the density of final states. To do this let the volume V be a cube of side L. We can impose periodic boundary conditions on the wave function, that is:
[tex]k_x=\frac{2\pi}{L}n_x[/tex]
[tex]k_y=\frac{2\pi}{L}n_y[/tex]
[tex]k_z=\frac{2\pi}{L}n_z[/tex]
where nx, ny and nz are positive or negative integers, or zero. Since L is very large we can treat nx, ny and nz as continuous variables, and the number of states in the range d[tex]\vec{k}=dk_xdk_ydk_z[/tex] is:
[tex]dn_xdn_ydn_z=\left(\frac{L}{2\pi}\right)^3dk_xdk_ydk_z=\left(\frac{L}{2\pi}\right)^3k^2dkd\Omega[/tex]

I can't understand the last equality, [tex]\Omega[/tex] is the solid angle, but how do I relate it to [tex]dk_xdk_ydk_z[/tex]?
 

Answers and Replies

  • #2
nicksauce
Science Advisor
Homework Helper
1,272
5
It is nothing tricky, it is just spherical coordinates.

[tex]d^3k = k^2dkd\Omega = k^2\sin{\theta}dkd\theta d\phi.[/tex]

If nothing depends on angle, then the solid angle can be integrated out to give a factor of 4pi.

(Edit: Actually you also need to divide by 8. All the n's are >0, whereas the factor of 4pi assumes that k can be negative and positive).
 
Last edited:
  • #3
200
1
Thank you!:smile:
 

Related Threads on Density of states in QM

  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
10
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
8
Views
883
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
7
Views
1K
Top