Prove Bloch's Theorem from Integral Schrodinger's Eqn

R}}\left[N e^{i \mathbf{k} \cdot \mathbf{r}} - \frac{m}{2 \hbar^2 \pi} \int d^3 r' \frac{\exp(\pm ik|\mathbf{r} - \mathbf{r}'|)}{|\mathbf{r}- \mathbf{r}'|}V(\mathbf{r}')\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}')\right]Now, we can see that the wave function on the right side of the equation is just \psi_{\mathbf{k}}^{(\pm)}(\mathbf{r
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king vitamin
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Prove Bloch's Theorem from Integral Schrodinger's equation (sorry! tried to make an umlaut and accidentally hit the keyboard shortcut for posting, any way to edit titles?)

Homework Statement



If a scattering potential has the translation invariance property [itex]V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})[/itex], where [itex]\mathbf{R}[/itex] is a constant vector, prove that the scattering solutions [itex]\psi_{\mathbf{k}}^{(\pm)}[/itex] of the integral form of the Schrodinger equation are Bloch wave functions, since they satisfy the relation

[itex]
\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}+\mathbf{R}) = e^{i \mathbf{k} \cdot \mathbf{R}}\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}).
[/itex]

Homework Equations



The integral form of Schrodinger's equation is given as:

[itex]
\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}) = N e^{i \mathbf{k} \cdot \mathbf{r}} - \frac{m}{2 \hbar^2 \pi} \int d^3 r' \frac{\exp(\pm ik|\mathbf{r} - \mathbf{r}'|)}{|\mathbf{r} - \mathbf{r}'|}V(\mathbf{r}')\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}')
[/itex]

The Attempt at a Solution



I'm don't seem to be getting anywhere here. I begin by computing:

[itex]
\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}+\mathbf{R}) = N e^{i \mathbf{k} \cdot (\mathbf{r}+\mathbf{R})} - \frac{m}{2 \hbar^2 \pi} \int d^3 r' \frac{\exp(\pm ik|\mathbf{r}+\mathbf{R} - \mathbf{r}'|)}{|\mathbf{r}+\mathbf{R} - \mathbf{r}'|}V(\mathbf{r}')\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}')
[/itex].

The first term is fine of course. In the second term, we obviously need to use the translation property of the potential, and we need to make the integral look like the original Schrodinger's equation for [itex]\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r})[/itex]. So I change variables [itex]\mathbf{r}' \rightarrow \mathbf{r}' + \mathbf{R}[/itex] (Jacobian is trivial for shifts):

[itex]
\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}+\mathbf{R}) = N e^{i \mathbf{k} \cdot (\mathbf{r}+\mathbf{R})} - \frac{m}{2 \hbar^2 \pi} \int d^3 r' \frac{\exp(\pm ik|\mathbf{r} - \mathbf{r}'|)}{|\mathbf{r}- \mathbf{r}'|}V(\mathbf{r}')\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}'+\mathbf{R})
[/itex]

where I have used the translation property of the potential to shift it. This is where I'm stuck. I can't really work with the exact wave function, but the equation as it is seems to be impossible to work with because the arguments of the Green's function and the wave function don't match. If I could even get a hint as to the idea behind this it would really help.

I can also write:

[itex]
\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}+\mathbf{R}) = e^{i \mathbf{k} \cdot \mathbf{R}} \left[N e^{i \mathbf{k} \cdot \mathbf{r}} - \frac{m}{2 \hbar^2 \pi} \int d^3 r' \frac{\exp(\pm ik|\mathbf{r} - \mathbf{r}'|)}{|\mathbf{r}- \mathbf{r}'|}V(\mathbf{r}')e^{-i \mathbf{k} \cdot \mathbf{R}}\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}'+\mathbf{R}) \right]
[/itex]
where I want to show
[itex]
e^{-i \mathbf{k} \cdot \mathbf{R}}\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}'+\mathbf{R}) = \psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}')
[/itex].
This would work if the wavefunction were a momentum eigenstate (replace k by p/hbar), but I don't think this is true in general (k is defined as [itex]k^2 = 2mE/\hbar^2[/itex]).
 
Last edited:
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  • #2





Thank you for your question. I would be happy to help you prove Bloch's Theorem from the integral form of Schrodinger's equation. First, I would like to point out that the translation property of the potential can be written as V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})e^{i \mathbf{k} \cdot \mathbf{R}}, where \mathbf{k} is a constant vector. This will be helpful in our proof.

Now, let's start by expanding the wave function \psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}+\mathbf{R}) as you have done:

\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}+\mathbf{R}) = N e^{i \mathbf{k} \cdot (\mathbf{r}+\mathbf{R})} - \frac{m}{2 \hbar^2 \pi} \int d^3 r' \frac{\exp(\pm ik|\mathbf{r} - \mathbf{r}'|)}{|\mathbf{r}- \mathbf{r}'|}V(\mathbf{r}')\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}'+\mathbf{R})

Next, we can use the translation property of the potential to rewrite the integral as:

\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}+\mathbf{R}) = N e^{i \mathbf{k} \cdot (\mathbf{r}+\mathbf{R})} - \frac{m}{2 \hbar^2 \pi} \int d^3 r' \frac{\exp(\pm ik|\mathbf{r} - \mathbf{r}'|)}{|\mathbf{r}- \mathbf{r}'|}V(\mathbf{r}')e^{i \mathbf{k} \cdot \mathbf{R}}\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}')

Now, we can rewrite the wave function as:

\psi_{\mathbf{k}}^{(\pm)}(\mathbf{r}+\mathbf{R}) = e^{i \mathbf{k
 

Related to Prove Bloch's Theorem from Integral Schrodinger's Eqn

1. What is Bloch's Theorem?

Bloch's Theorem is a fundamental principle in solid-state physics that describes the behavior of electrons in a periodic crystal lattice. It states that the wavefunction of an electron in a crystalline solid can be written as a product of a periodic function and a plane wave.

2. What is the significance of Bloch's Theorem?

Bloch's Theorem allows us to simplify the description of electrons in a crystal, making it easier to understand and predict their behavior. It also forms the basis for many other important principles in solid-state physics, such as band theory and the concept of energy bands.

3. What is the relationship between Bloch's Theorem and Schrodinger's Equation?

Bloch's Theorem can be derived from the integral form of Schrodinger's Equation, which describes the time evolution of quantum systems. By applying Bloch's Theorem to this equation, we can obtain a simpler form that is specifically applicable to electrons in a crystal lattice.

4. How is Bloch's Theorem proven from Integral Schrodinger's Equation?

The proof involves applying Bloch's Theorem to the integral form of Schrodinger's Equation and using the periodicity of the crystal lattice to simplify the equation. This results in a simplified equation that describes the behavior of electrons in a crystal lattice, which is known as the Bloch Equation.

5. What are some real-world applications of Bloch's Theorem?

Bloch's Theorem is essential for understanding and predicting the electronic properties of materials, which has numerous practical applications. For example, it is used in the design of electronic devices such as transistors and lasers, as well as in the development of new materials for renewable energy technologies.

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