PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
Felix Bloch showed that the solutions to the Schrödinger equation for an electron in a periodic potential 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. Solutions of this form are eigenfunctions of the translation operator $\mathbf{T}$. This is easily demonstrated by shifting the position of 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 eigen vector $e^{i\vec{k}\cdot\vec{T}}$.
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.