Let \(B>0\). The magnetic derivatives at magnetic strength \(B\) are defined as
\[ \widetilde{\partial }_1=i\partial_1 - \frac{B}{2}x_2 \quad \text{and} \quad \widetilde{\partial }_2=i\partial_2 + \frac{B}{2}x_1. \]
Remark
- The Landau operator \(H_B\) may be expressed in terms of magnetic derivatives as a Laplacian, that is \(H_B=\widetilde{\partial }_1^2+\widetilde{\partial }_2^2\).
- Both derivatives do not commute, but following relation hold \([\widetilde{\partial }_1, \widetilde{\partial }_2]=iB\).
- There is a relation between ordinary and magnetic derivatives, namely: \[ i\partial_i (u\bar{v}) = \bar{v} \widetilde{\partial}_iu - u \overline{\widetilde{\partial}_i v}.\]