# Particle in a Magnetic Field: Momentum Representation of the Hamiltonian

We consider a particle in a magnetic field $$\vec{B}$$, which derives from the vector potential $$\vec{A}$$ by $$\vec{B} = \nabla \times \vec{A}$$. The Schrödinger equation in position representation reads

$$\hat{H}_{\vec{k}} \, u_{n, \vec{k}} = \frac{\hbar^2}{2m} \left(-i \nabla + \vec{k} – \vec{A} \right)^2 u_{n, \vec{k}} = \varepsilon_{n, \vec{k}} \, u_{n, \vec{k}}$$

for the Bloch wavefunction $$\psi_{n, \vec{k}}(\vec{r}) = e^{i \vec{k} \cdot \vec{r}} u_{n, \vec{k}}(\vec{r})$$. Next, we employ rationalized units $$\hbar = c = e = 1$$ and expand the Hamiltonian

$$\hat{H}_{\vec{k}} u_{n, \vec{k}} = \frac{1}{2} \left( – \Delta u_{n, \vec{k}} + 2 i (\vec{A} – \vec{k}) \cdot \nabla u_{n, \vec{k}} + (\vec{A} – \vec{k})^2 u_{n, \vec{k}} \right)$$

The momentum representation is derived by inserting a “one” composed of position eigenstates as we obtained already position representation:

$$\langle k | \hat{H}_{\vec{k}} | n, \vec{k} \rangle = \int d^2{r} \langle k | r \rangle \langle r | \hat{H}_{\vec{k}} | n, \vec{k} \rangle$$

$$= \frac{1}{2} \frac{1}{S} \int d^2{r} \, e^{-i \vec{g} \cdot \vec{r}} \left[ – \Delta u_{n, \vec{k}} + 2 i (\vec{A} – \vec{k}) \cdot \nabla u_{n, \vec{k}} + (\vec{A} – \vec{k})^2 u_{n, \vec{k}} \right]$$

\begin{aligned} = \frac{1}{2} & [ \vec{g}^2 u_{n, \vec{k}}(\vec{g}) \delta_{\vec{g},\vec{g}^\prime} + 2 (-\vec{g}) \vec{A}(\vec{g} -\vec{g}^\prime) u_{n, \vec{k}}(\vec{g}) + 2 \vec{k} \cdot \vec{g} u_{n, \vec{k}}(\vec{g}) \delta_{\vec{g},\vec{g}^\prime} \\ & + \vec{A}^2(\vec{g} -\vec{g}^\prime) u_{n, \vec{k}}(\vec{g}) – 2 \vec{A}(\vec{g} -\vec{g}^\prime) \cdot \vec{k} u_{n, \vec{k}}(\vec{g}) + \vec{k}^2 u_{n, \vec{k}}(\vec{g}) \delta_{\vec{g},\vec{g}^\prime} ] \end{aligned}

\begin{aligned} = \frac{1}{2} & [ (\vec{k}^2 + \vec{g}^2) u_{n, \vec{k}}(\vec{g}) \delta_{\vec{g},\vec{g}^\prime} – 2 (\vec{g}) \vec{A}(\vec{g} -\vec{g}^\prime) \cdot (\vec{k} + \vec{g}) u_{n, \vec{k}}(\vec{g}) \\ & + \vec{A}^2(\vec{g} -\vec{g}^\prime) u_{n, \vec{k}}(\vec{g}) ] \end{aligned}

with the convolution

$$\vec{A}^2(\vec{r}) = \sum \vec{A}(\vec{g} -\vec{g}^\prime) \cdot \vec{A}(\vec{g}^\prime)$$

So the Hamiltonian in momentum representation is given by

$$\hat{H}_{\vec{k}}(\vec{g},\vec{g}^\prime) = \frac{1}{2} \left[ (\vec{k}^2 + \vec{g}^2) \delta_{\vec{g},\vec{g}^\prime} – 2 (\vec{g}) \vec{A}(\vec{g} -\vec{g}^\prime) \cdot (\vec{k} + \vec{g}) + \vec{A}^2(\vec{g} -\vec{g}^\prime) \right]$$

in the Coulomb gauge $$\nabla \cdot \vec{A} = 0$$ and $$\vec{g} \cdot \vec{A} = 0$$, respectively.

Taillefumier, Mathieu et al. “Chiral two-dimensional electron gas in a periodic magnetic field: Persistent current and quantized anomalous Hall effect.” Physical Review B 78 (2008): 155330.
https://arxiv.org/abs/0807.0707v1