Magnetic translation

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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  (R_x,R_y) .

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  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
 e^{i(p_x X + p_y Y)},
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.

References[edit]

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