Magnetic translation

Magnetic translation

Magnetic translations are naturally defined operators acting on wave function on a two-dimensional particle in a magnetic field.

According to [1], the motion of an electron in a magnetic field on a plane is described by the following four variables: guiding center coordinates (X,Y) and the relative coordinates (Rx,Ry).

The guiding center coordinates are independent of the relative coordinates and, when quantized, satisfy
 [X,Y]=-i \ell_B^2 ,
where  \ell_B=\sqrt{\hbar/eB} , which makes them mathematically similar to the position and momentum operators Q = q and  P=-i\hbar \frac{d}{dq} in one-dimensional quantum mechanics.

Much like acting on a wave function f(q) of a one-dimensional quantum particle by the operators eiaP and eibQ generate the shift of momentum or position of the particle, for the quantum particle in 2D in magnetic field one considers the magnetic translation operators
 e^{i(p_x X + p_y Y)},
for any pair of numbers (px,py).

The magnetic translation operators corresponding to two different pairs (px,py) and (p'x,p'y) do not commute.

References

  1. ^ Z.Ezawa. Quantum Hall Effect, 2nd ed, World Scientific. Chapter 28

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