PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
Felix Bloch considered the to the Schrödinger equation for an electron in a periodic potential,
$$-\frac{\hbar^2}{2m}\nabla^2\psi + V(\vec{r})\psi = E\psi,$$and showed that the solutions have the form,
$$\psi_{\vec{k}}(\vec{r}) = e^{i\vec{k}\cdot\vec{r}}u_{\vec{k}}(\vec{r}),$$where $u_{\vec{k}}(\vec{r})$ is a periodic function with the periodicity of the Bravais lattice and $\vec{k}$ is a wave vector in the first Brillouin zone. Any wavefunction that satisfies periodic boundary conditions can be written as a sum of Bloch wavefunctions. Consider an arbitrary wavefunction that satisfies periodic boundary conditions,
$$\psi(\vec{r})=\sum\limits_{\vec{k}}C_{\vec{k}}e^{i\vec{k}\cdot\vec{r}}.$$The sum over all $\vec{k}$ can be written as a double sum over $\vec{k}$ in the first Brillouin zone and a sum over the reciprocal lattice vectors,
$$\psi(\vec{r})=\sum\limits_{\vec{k}\in \text{1BZ}}\sum\limits_{\vec{G}}C_{\vec{k}+\vec{G}}e^{i(\vec{k}+\vec{G})\cdot\vec{r}}=\sum\limits_{\vec{k}\in \text{1BZ}}e^{i\vec{k}\cdot\vec{r}}\sum\limits_{\vec{G}}C_{\vec{k}+\vec{G}}e^{i(\vec{G})\cdot\vec{r}}.$$Since $u_{\vec{k}+\vec{G}}(\vec{r}) = \sum\limits_{\vec{G}}C_{\vec{k}+\vec{G}}e^{i(\vec{G})\cdot\vec{r}}$ is a periodic function, any periodic wavefucntion can be written as a sum of Bloch wavefunctions,
$$\psi(\vec{r})=\sum\limits_{\vec{k}\in \text{1BZ}}e^{i\vec{k}\cdot\vec{r}}u_{\vec{k}+\vec{G}}(\vec{r}).$$Solutions of this form are eigenfunctions of the translation operator $\mathbf{T}$. This is easily demonstrated by shifting the position of the wave function by a translation vector of the Bravais lattice $\vec{T} = h\vec{a}_1 +k\vec{a}_2 +l\vec{a}_3$. Here $\vec{a}_1$, $\vec{a}_2$, and $\vec{a}_3$ are the primitive lattice vectors of the Bravais lattice and $h$, $k$, and $l$ are integers. Applying the translation operator to the wavefunction $\psi_{\vec{k}}(\vec{r})$ yields,
$$\mathbf{T}\psi_{\vec{k}}(\vec{r})= e^{i\vec{k}\cdot(\vec{r}+\vec{T})}u_{\vec{k}}(\vec{r}+\vec{T})$$Since $u_{\vec{k}}(\vec{r})$ is a periodic function, it is unchanged by this translation,
$$\mathbf{T}\psi_{\vec{k}}(\vec{r})=e^{i\vec{k}\cdot\vec{T}}e^{i\vec{k}\cdot\vec{r}}u_{\vec{k}}(\vec{r})= e^{i\vec{k}\cdot\vec{T}}\psi_{\vec{k}}(\vec{r})$$Therefore $\psi_{\vec{k}}(\vec{r})$ is an eigenfunction of the translation operator with an eigenvalue $e^{i\vec{k}\cdot\vec{T}}$.
Notice that the electron probability density,
$$\psi^*_{\vec{k}}(\vec{r})\psi_{\vec{k}}(\vec{r}) = e^{-i\vec{k}\cdot\vec{r}}u^*_{\vec{k}}(\vec{r})e^{i\vec{k}\cdot\vec{r}}u_{\vec{k}}(\vec{r})= u^*_{\vec{k}}(\vec{r})u_{\vec{k}}(\vec{r}),$$is the product of two periodic functions and is also periodic with the same periodicity.
If the potential that the electron sees is periodic, then the Hamiltonian will commute with the translation operator, and it will be possible to find eigenfunctions that are simultaneously eigenfunctions of the translation operator and eigenfunctions of the Hamiltonian. One way to show that solutions of Bloch form solve the Schrödinger equation for electrons in a periodic potential is to assume this form and substitute it into the Schrödinger equation. This is done in the section on the plane wave method. Solutions of Bloch form typically do not have a simple analytic form. They have to be calculated numerically. The next section discusses numerical calculations of the Bloch solutions for one-dimensional potentials.