Proving Bloch's Theorem

  • Thread starter naele
  • Start date
  • Tags
    Theorem
In summary, the conversation discusses how to show that a Hamiltonian with a periodic potential commutes with the translation operator. The key is to use the definition of how the translation operator acts on a ket and consider the traditional function notation for the equation.
  • #1
naele
202
1
One of the more common ways of showing that a Hamiltonian with periodic potential commutes with the translation operator is to write the following (like Ashcroft and Mermin p. 133)

[tex]
T(R)H(r)\psi(r)=H(r+R)\psi(r+R)=H(r)T(R)\psi(r)
[/tex]

I suspect this might be a dumb question, but what allows us to write [itex]T(R)H(r)\psi(r)=H(r+R)\psi(r+R)[/itex], that is why is the translation operator acting on both the Hamiltonian and the wave, and not just on the Hamiltonian?
 
Physics news on Phys.org
  • #2
I think its because the potential is periodic then the Hamiltonian is too: H(x)=H(x+a), you can then sub this in directly and the translation operator now just acts on psi
 
  • #3
naele said:
I suspect this might be a dumb question, but what allows us to write [itex]T(R)H(r)\psi(r)=H(r+R)\psi(r+R)[/itex]
Because that is the definition of how the space translation operator acts on a ket.

It may help to write [itex]\theta(r) = H(r) \psi(r)[/itex]. [itex]\theta(r)[/itex] is a ket. What [itex]T(R) \theta(r)[/itex]...It may help more to consider more traditional function notation for what I believe is being written:
[tex] (T(R) H \psi)(r) = (H \psi)(r + R).[/tex]
 

1. What is Bloch's Theorem?

Bloch's Theorem is a fundamental principle in solid state physics that states that the wave function of an electron in a crystal lattice has the same periodicity as the lattice itself, and can be expressed as a product of a periodic function and a plane wave.

2. Why is Bloch's Theorem important?

Bloch's Theorem allows for a simplified description of the electronic structure of crystalline materials, which is crucial in understanding their physical and chemical properties. It also serves as the basis for many other important concepts in solid state physics, such as band theory and the concept of a Brillouin zone.

3. How is Bloch's Theorem proven?

Bloch's Theorem can be proven using mathematical techniques such as Fourier analysis and perturbation theory. It can also be derived from more general principles such as the Schrödinger equation and the translation symmetry of the crystal lattice.

4. Are there any limitations to Bloch's Theorem?

While Bloch's Theorem is a powerful tool in describing the electronic structure of perfect crystalline materials, it does have limitations. It does not hold for materials with disordered or amorphous structures, and it does not take into account the effects of electron-electron interactions.

5. How does Bloch's Theorem relate to experimental data?

Bloch's Theorem provides a theoretical framework for understanding the electronic structure of crystalline materials, but experimental data is still needed to validate its predictions. Various experimental techniques, such as X-ray diffraction and photoemission spectroscopy, can be used to measure the properties of materials and compare them to the predictions of Bloch's Theorem.

Similar threads

Replies
9
Views
359
  • Quantum Physics
Replies
4
Views
726
Replies
1
Views
647
  • Quantum Physics
Replies
1
Views
530
Replies
4
Views
1K
Replies
1
Views
591
Replies
3
Views
777
Replies
1
Views
703
  • Quantum Physics
Replies
14
Views
829
  • Atomic and Condensed Matter
Replies
5
Views
1K
Back
Top