I'm confused


by Petar Mali
Tags: confused
Petar Mali
Petar Mali is offline
#1
Jan25-11, 01:01 PM
P: 290
If I have cubic structure where plane is define by vector [tex]\rho[/tex] and in [tex]z[/tex] direction I have planes [tex]...m-2,m-1,m,m+1,m+2...[/tex]

and if I have for example

[tex]\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m+1,\vec{\rho}}[/tex]

how to go with that in K-space?

If I had


[tex]\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}[/tex]

I will say

[tex](m,\vec{\rho})=\vec{n}[/tex]

and then I will have


[tex]\sum_{\vec{n}}\hat{B}_{\vec{n}}\hat{B}_{\vec{n}}=\sum_{\vec{n}}\frac{1} {\sqrt{N}}\sum_{\vec{k}}\hat{B}_{\vec{k}}e^{i\vec{k}\cdot\vec{n}}\frac{ 1}{\sqrt{N}}\sum_{\vec{q}}\hat{B}_{\vec{q}}e^{i\vec{q}\cdot\vec{n}}[/tex]

[tex]=\frac{1}{N}\sum_{\vec{k},\vec{q}}\hat{B}_{\vec{k}}\hat{B}_{\vec{q}}N\d elta_{\vec{k},-\vec{q}}=\sum_{\vec{k}}\hat{B}_{\vec{k}}\hat{B}_{-\vec{k}}[/tex]

But what to do in case with m+1. Thanks for your answer!
Phys.Org News Partner Physics news on Phys.org
Researchers find tin selenide shows promise for efficiently converting waste heat into electrical energy
After 13 years, progress in pitch-drop experiment (w/ video)
Global scientific team 'visualizes' a new crystallization process (w/ video)
shawl
shawl is offline
#2
Jan26-11, 11:08 PM
P: 63
I am not familiar with these equations, but I try to give some opinions for discussion.

The problem is that there will be an unexpected term, i.e. exp(i*q), if the B(m)B(m+1) are transformed in K-space.

As I learned, B(m)B(m+1) denotes the transition process between state m and state m+1.

The complete formula is usually written in the sum of V(m,m+1)B(m)B(m+1), where V(m,m+1) is the transition matrix element.

When the formula is transformed in K-space, V(m,m+1) are also transformed as V(m,k,m+1,q), or written in V(k,q) for the shortness.

And what i am thinking is that the unexpected term exp(i*q) will be absorbed in V(k,q).
That means you can do the transformation in the case of (m,m+1) just like what you did in the case of (m,m). The difference for the m and m+1 only appears in the transition matrix elements.
kanato
kanato is offline
#3
Jan27-11, 01:17 AM
P: 416
Yes, as shawl mentions you get an extra factor of exp(i*q). With the right symmetry in your lattice you will be able to combine the exponentials to end up with something like [tex]\sum_q f(q) B_q B_{-q}[/tex] where f(q) is some real function, probably composed of cosines. This is exactly the sort of thing you get in tight binding, except there you have a creation and annihilation operator on different sites.


Register to reply

Related Discussions
Confused !!!! General Physics 7
Little confused by this? Calculus & Beyond Homework 2
little confused Calculus & Beyond Homework 2
a little confused Atomic, Solid State, Comp. Physics 14
3x + 1 - x - 1 = 2 General Math 12