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I am reading the following review paper on the Quantum Hall Effect. I am sorry for the extremely stupid question, but I have been stuck on this very easy equation for long.

In equation 2.39, the author derives the following commutation relation between the coordinates of the center of a Landau Level.


$l_B$ called as the magnetic length is $\sqrt{\frac{\hbar}{eB}}$. From this equation, the author says using the uncertainty relation(eqn 2.40)

$$\Delta X \Delta Y = 2 \pi l_B^2$$

How do you get this equation. I know that the general uncertainty relation equality is $\Delta A^2 \Delta B^2=<\frac{1}{2i}[A,B]>^2$, but this is obviously not giving the required answer. $\frac{1}{2i}il_B^2=l_B^2/2=\frac{h}{4 \pi eB}$. Why is this not the correct answer? Have they used some stronger form of the uncertainty principle?

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Hm. The only things I can think of are a) the author is using a slightly different definition of the uncertainty $\Delta X$, i.e. not the standard deviation of $X$, but something like the FWHM or something proportional to it, or b) there is a stronger uncertainty principle - the Schrodinger uncertainty principle, Eq. 3 of cds.cern.ch/record/499991/files/0105035.pdf - but it's only a different bound if the covariance of X and Y is non-zero, and I don't know if that is true in your case. Since the paper doesn't work out the anticommutator, I'd bet on a. –  AJK Apr 22 '13 at 6:58
Could just be theory units: constants of order unity = 1 –  wsc Apr 22 '13 at 14:30
@AJK ,wsc: I have a feeling that this has something to do with the semi-classical quantization of phase space. When we divide phase space into cells, why is there an extra factor of $2 \pi$? I remember coming across something of this sort some months back. I will try to study some phase space quantization to be able to explain this. But do you have anything to say about this? –  ramanujan_dirac Apr 22 '13 at 16:14

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The uncertainty principle in use here is not the usual Heisenberg uncertainty principle, but the semi-classical quantization of phase space. If two operators have the relation $[X,Y]=ik$, then the minimal area in phase space is $2 \pi k$. For more info: How does one quantize the phase-space semiclassically?

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