Magnetic translation explained

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

The motion of an electron in a magnetic field on a plane is described by the following four variables:^[1] guiding center coordinates

(X,Y)

and the relative coordinates

(R_x,R_y)

The guiding center coordinates are independent of the relative coordinates and, when quantized, satisfy

[X,Y]=-i

	2
\ell
	B

,
where

\ell_{B=\sqrt{\hbar/eB}}

, which makes them mathematically similar to the position and momentum operators

Q=q

and

P=-i\hbar

	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

e^iaP

and

e^ibQ

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

	i(p_xX+p_yY)
e

for any pair of numbers

(p_x,p_y)

The magnetic translation operators corresponding to two different pairs

(p_x,p_y)

and

(p'_x,p'_y)

do not commute.

Notes and References

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