Theory to argue whether matrix elements vanish or not?

In summary, the conversation discusses the use of a Hamiltonian, complete basis set, and symmetric operators to calculate matrix elements at various crystal momentum points. The speaker is curious about the physical meaning behind the resulting zeros and non-zeros at different k points and is seeking clarification on how group theory can help in understanding these results. A helpful reference is also provided for further study.
  • #1
Hao Hsu
2
0
I have known:
(1) A Hamiltonian, say, H(k), where k is the crystal momentum.
(2) An appropriate complete basis set {a_1,a_2,a_3…}.
(3) Some symmetric operators {A,B,…} which commute with H(k), i.e. [A,H]=[B,H]=...=0.

Of course, by calculation, I can get any matrix element of H(k), i.e. <a_i|H(k)|a_j>.

After numerical calculation, not that surprisingly, I get many zeros of matrix elements at high symmetric k points. However, is there any method for me to verify the results or discuss the physical meaning?

For example, why for
(1) k=Γ point: <a_1|H(k=Γ)|a_3>=0 and <a_1|H(k=Γ)|a_5>≠0
(2) k=X point: <a_1|H(k=X)|a_3>≠0 and <a_1|H(k=X)|a_5>≠0?
Why there exists difference between different k points?

I think those symmetric operators might provide some help, but I am not familiar with this approach. Or are there some other techniques which might do the job for me? Thank you very much guys for reading my question!
 
Last edited:
Physics news on Phys.org
  • #2
That's where you need group theory! I.e. you can determine the irreducible presentation each operator belongs to under the given little group corresponding to the specified symmetry point. You can do so also for your states a_i. Then you can also determine which irreducible representations are present in the decomposition the product of the states with the operator. The product has to contain the totally symmetric representation for the matrix element to be non-zero.
Here is some nice reference:
http://web.mit.edu/course/6/6.734j/www/group-full02.pdf
 
  • Like
Likes Hao Hsu and DrClaude
  • #3
Thank you very much! I will study them this weekend ^^
 

1. What is the theory behind matrix elements?

Theory of matrix elements is a mathematical concept used in quantum mechanics to describe the transition between two states of a quantum system. It involves calculating the probability of a particle transitioning from one state to another, and is based on the principles of linear algebra.

2. How do we determine if matrix elements vanish or not?

The value of matrix elements can be determined by solving the Schrödinger equation for the quantum system in question. If the resulting value is zero, then the matrix element vanishes. This is often used to determine the allowed energy levels of a system.

3. What is the significance of matrix elements in quantum mechanics?

Matrix elements play a crucial role in understanding the behavior of quantum systems. They provide a way to calculate the probabilities of different outcomes and to predict the behavior of particles in a quantum system. They also help in understanding the energy levels of a system and the transitions between them.

4. Can matrix elements be negative?

Yes, matrix elements can have negative values. This is because they are derived from complex numbers and can have both real and imaginary components. The negative values can also indicate a phase difference between the initial and final states of a quantum system.

5. How are matrix elements related to observable quantities?

In quantum mechanics, observable quantities such as energy, position, and momentum are represented by operators. These operators act on the wave function of a system, and the matrix elements of these operators represent the values of the corresponding observable quantities. Therefore, matrix elements provide a way to connect theoretical concepts to observable phenomena.

Similar threads

  • Atomic and Condensed Matter
Replies
3
Views
733
  • Atomic and Condensed Matter
Replies
1
Views
813
  • Atomic and Condensed Matter
Replies
7
Views
2K
  • Atomic and Condensed Matter
Replies
0
Views
247
  • Atomic and Condensed Matter
Replies
5
Views
2K
  • Atomic and Condensed Matter
Replies
2
Views
3K
Replies
4
Views
726
  • Introductory Physics Homework Help
Replies
8
Views
798
  • Atomic and Condensed Matter
Replies
2
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
694
Back
Top