As quantum-mechanical objects, electrons and holes do not have specific coordinates; they are described in terms of wave functions whose absolute squared in the coordinate representation describe the probability distributions of these being anywhere in the underlying configuration space. By including interaction between electrons and holes (as in the case of describing excitons, which are bound states of electrons and holes -- bound states come in a variety of forms, so that one can for instance have biexcitons, molecules each consisting of two exitons), one has to do with a joint probability distribution that depends on the coordinates associated with electrons and holes in the coordinate representation. Evidently, the form of this joint probability distribution function depends amongst others on the type of the interaction between electrons and holes. By considering a rotationally symmetric hard-core interaction potential, this probability distribution will be identically vanishing for the distance between the two coordinates being less than twice the core radius of the potential.

However, if a two-dimensional semiconductor with limited linear dimensions L_x and L_y is placed in an external uniform magnetic field B, then the conducting electrons and holes begin to move along quantum orbits that have centers with the y_0=p_x c/eB coordinate that commutes with the Hamiltonian (see page 425 in [1] (attached)), that is, y_0 is conserved with time. The quantity y_0 is degenerate in spin, that is, two identical electrons with opposite spin projections (on the vector B) can have one value of the coordinate of the center of the orbit y_0. The value of y_0 is determined by the x-projection of its momentum p_x=h2\pi/L_x. Therefore seems that for the same size L_x of the sample, the value y_0 for the hole will be the same. That is why I asked the question:

Can an electron and a hole have one common coordinate y_0 of the center of the quantum orbit in a two-dimensional semiconductor in a magnetic field?

[1] Landau Quantum Mechanics. Non-relativistic Theory (attached here)

Vladimir V. Lugovoi : Whether x0 or y0, or a combination of the two, is a conserved quantity, depends entirely on the choice of the gauge, which is not unique [1].

Be it as it may, the issue regarding y0 being or no being a conserved quantity (commuting with the Hamiltonian) has not been part of the question posed on this page. Further, since you are referring to the book by L&L as well as x0 and y0, you must now be focusing on a system of two-dimensional electrons, or two-dimensional holes, while in the main question you refer to a system of electrons and holes, for which one is inclined to think about a bilayer system, with electrons residing on one layer and holes on the other. If we therefore (conform your above response) focus on a system of two-dimensional electrons in the quantum Hall regime, and using the Landau gauge, A = (-H y, 0, 0), then the eigenstates corresponding to a given Landau level are degenerate, the eigenvalues do not depending on px (momentum in the x direction). This is relevant to the question uder discussion in that for two independent eigenstates corresponding to the same Landau level the corresponding px's are necessarily different, and importantly y0 linearly depends on px, implying that the corresponding y0's must be different (even by an infinitesimal amount for macroscopic systems).

[1] The gauge non-invariance of x0 and y0 implies that these quantities are not measurable.

Dear Behnam Farid ,

Let me clarify my doubts.

Landau comments on the formula (112.4) and writes:

"The eigenvalues p_x and p_z take all values from minus infinity to plus infinity. "

Further.

The value y_0 for the electron (e = - | e | and if p_z> 0)

coincides with

the value of y_0 for the positron (e = | e | and if p_z

Vladimir V. Lugovoi : I think you are conflating unrelated issues (I cannot for instance see why holes and positrons should be relevant to the discussion at all). As I mentioned earlier, x0 and y0 are not gauge-invariant quantities, so that they cannot be assigned fixed values (they are not measurable quantities).

We regard to the statement

"The eigenvalues p_x and p_z take all values from minus infinity to plus infinity."

to which you refer, this has to do with the uniformity of the system of charged particles (the sign of the charge is immaterial for the discussion at hand) in the absence of an external magnetic field, subject to the box boundary condition, with the size of the box assumed to be macroscopically large. Were it not for the magnetic field, the eigenfunctions of the underlying Schrödinger equation would be plane waves, characterised by linear momenta px, py, and pz. Applying a constant magnetic field in the z-direction, for the relevant vector potential in the Landau gauge one has A = (-H y, 0, 0). This leads, through the covariant momentum operator, to a Schrödinger equation that remains translational invariant in the x- and z-directions, but no longer in the y-direction. That is why in this case px and pz remain good quantum numbers; thus, as in the case of the zero magnetic field and under the box boundary condition, with the box size macroscopically large, px and pz take values over the entire real axis.

Dear Behnam Farid ,

You're right. But further. When a magnetic field appears along the z axis (in the Landau gauge), electrons begin to move on cyclic orbits with coordinate centre y_0, which in [1] is called “a good quantum number”, which allows calculating the number of states per unit area (see [1] and page 426 in [2]) in sample with (micro) dimensions L_x, L_y. Therefore, in our calculations, we believe that, knowing L_x, L_y, we can calculate the “good quantum number” y_0 - the coordinate of each center of the orbits in the area of ​​the sample, which, in the one case, can contain only holes or only electrons, or both , and another, as it is in graphene. In this scheme, we begin to calculate the interaction energies. Of course, it is necessary to take into account the distortions introduced by the distance between the layers with hole and electron conductivity. But the layers can be combined as it is in graphene. In any way , holes and electrons will be participate in process of interaction for different y_0.

[1]- Klaus von Klitzing, Nobel lecture, December 9, 1985

[2]- Landau, Lifshitz Quantum Mechanics. Non-relativistic Theory.

Vladimir V. Lugovoi : As regards y0 being a "good quantum number", it is not; the degeneracy factor N (see Eq. (9) in the Nobel Lecture by von Klitzing, to which you refer) is in contrast a good quantum number -- it is determined from the gauge-invariant quantity Δy0 as defined in Eq. (8) of the latter Nobel Lecture (note that for the N0 in the latter equation one has N0 = Ly/Δy0). For using the term "good quantum number", von Klitzing refers to the 1982 publication by Laughlin in the journal Surface Science [1], but in this publication the term "good quantum number" is not used at all. Incidentally, a quantity like y0 being gauge non-invariant does not mean that one cannot construct a function of it that is well gauge invariant. A case in point is the above-mentioned gauge-invariant Landau-level degeneracy factor N.

As for the rest, I think we have already overdone this problem.

[1] RB Laughlin, Surf. Sci. 113, 22 (1982).

Article Impurities and edges in the quantum Hall effect

Dear Behnam farid,

thank you for expressing your point of view.

Vladimir