Question

According to the Bloch theorem, which of the following equations is the correct form for Bloch functions:

• A. \( \psi(x) = e^{\pm ikx} + u_k(x) \)
• B. \( \psi(x) = e^{\pm ikx} - u_k(x) \)
• C. \( \Psi(x) = e^{\pm ikx \cdot r} u_k(x,r) \)
• D. \( \psi(\mathbf{r}) = e^{\pm i\mathbf{k} \cdot \mathbf{r}} u_k(\mathbf{r}) \)

Hint:

Remember Bloch's theorem as "plane wave times a periodic part". This simple description helps you immediately identify the correct mathematical form, which is always a product, not a sum or difference.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Bloch's theorem is a fundamental theorem in solid-state physics that describes the form of the wave function for an electron in a periodic potential, such as the potential created by the crystal lattice.
Step 2: Key Formula or Approach:
Bloch's theorem states that the eigenfunctions of the Schrödinger equation for a periodic potential can be written as the product of a plane wave \(e^{i\mathbf{k} \cdot \mathbf{r}}\) and a function \(u_k(\mathbf{r})\) that has the same periodicity as the crystal lattice.
The mathematical form is:
\[ \psi_k(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} u_k(\mathbf{r}) \]
where \(u_k(\mathbf{r})\) is a periodic function such that \(u_k(\mathbf{r} + \mathbf{R}) = u_k(\mathbf{r})\) for all lattice vectors \(\mathbf{R}\).
Step 3: Detailed Explanation:
Let's analyze the given options based on the correct form of the Bloch function:


(A) and (B) show a sum or difference, which is incorrect. The theorem specifies a product.
(C) has inconsistent notation, mixing 1D and 3D variables (\(x\) and \(\mathbf{r}\)).
(D) shows the correct form: a plane wave part multiplied by a periodic part. Although the original option has a typo (\(\psi(x) = e^{\pm ikx \cdot r} u_k(x)\)), it is the only option that represents the product structure of a plane wave and a periodic function. We have presented the corrected 3D form, which is the general statement of the theorem. The \( \pm \) sign in the options simply indicates that waves can travel in either direction.

Step 4: Final Answer:
The correct form of a Bloch function is a plane wave modulated by a function that has the periodicity of the crystal lattice.