Thread Closed

eigen state

 
Share Thread Thread Tools
Jun20-05, 08:46 AM   #1
 
Thumbs down

eigen state


Consider particle of mass m in a cubic box of length L which has energy spectrum given by E=(k sqr)/2m =2 (pi sqr) (nx sqr+ ny sqr +nz sqr)/m (L sqr).what will be the density of states (eigen states per unit energy interval)

k is boltzman const..nx,ny,nz are unit vectors in resp. directions....
 
PhysOrg.com
PhysOrg
science news on PhysOrg.com

>> Hong Kong launches first electric taxis
>> Morocco to harness the wind in energy hunt
>> Galaxy's Ring of Fire
Jun20-05, 11:41 AM   #2
 
Recognitions:
Homework Helper Homework Help
Science Advisor Science Advisor
Quote by aura
Consider particle of mass m in a cubic box of length L which has energy spectrum given by E=(k sqr)/2m =2 (pi sqr) (nx sqr+ ny sqr +nz sqr)/m (L sqr).what will be the density of states (eigen states per unit energy interval)

k is boltzman const..nx,ny,nz are unit vectors in resp. directions....
Aren't the n values the eigenstates with n_j = 1,2,3....?

This looks to be a counting problem to find the number of combinations of the three n values leading to the same sum of squares. Clearly, for the lowest energy there is only one. After that, what are the possibilities, and what happens to the difference between energy levels as the n values increase?
 
Jun20-05, 12:22 PM   #3
 
Quote by OlderDan
Aren't the n values the eigenstates with n_j = 1,2,3....?

This looks to be a counting problem to find the number of combinations of the three n values leading to the same sum of squares. Clearly, for the lowest energy there is only one. After that, what are the possibilities, and what happens to the difference between energy levels as the n values increase?
oops! a big printing mistake...thats eigen vector...nx,ny,nz

now can u solve this at least the explanation...
 
Jun20-05, 11:56 PM   #4
 
Recognitions:
Homework Helper Homework Help
Science Advisor Science Advisor

eigen state


Quote by aura
oops! a big printing mistake...thats eigen vector...nx,ny,nz

now can u solve this at least the explanation...
[tex] E_{n_x,n_y,n_z} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[n_x^2+n_y^2+n_z^2\right] [/tex]

[tex] E_{1,1,1} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[3\right] [/tex]

[tex] E_{2,1,1} = E_{1,2,1} = E_{1,1,2} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[6\right] [/tex]

[tex] E_{2,2,1} = E_{2,1,2} = E_{1,2,2} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[9\right] [/tex]

[tex] E_{3,1,1} = E_{1,3,1} = E_{1,1,3} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[11\right] [/tex]

[tex] E_{2,2,2} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[12\right] [/tex]

[tex] E_{1,2,3} = E_{1,3,2} = E_{2,1,3} = E_{2,3,1} = E_{3,1,2} = E_{3,2,1} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[14\right] [/tex]

etc, etc.

I believe you are supposed to be figuring out all possible energies and how many degenerate states there are for each energy, and then divide the number of states by some energy interval to find the density. Unless I have missed some, the density is a bit erratic for these low numbered states. For larger n, perhaps you can come up with a general expression for how many states there are between some energy E and and a slightly higher level to come up with a number of states per unit energy interval. The sum of squares is suggestive that thinking in terms of the number of states contained within a spherical energy surface might prove helpful.
 
Jun21-05, 05:55 AM   #5
 
thanks for trying it out ...

i will try to solve it...


thanks!!
 
Thread Closed
Thread Tools


Similar Threads for: eigen state
Thread Forum Replies
eigen value Q. pls help General Math 6
figuring out if the state x is an eigen state of the hamiltonian Advanced Physics Homework 13
eigen values General Math 6
Eigen question Calculus & Beyond Homework 14
An eigen-What? Calculus & Beyond Homework 2