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arXiv:2606.13290 · International Journal of Theoretical Physics

Understanding quantum behaviors of an electron in a uniform magnetic field alternatively

Jin-Ming Wang, Yuan-Zao Gao, Dai-Lin Cun, Jian Jing

cond-mat.mes-hallquant-phSubmitted Jun 11, 2026• Announced Jun 12, 2026• 0 citations
v1 · Jun 11, 2026v2 · Jun 30, 2026
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Abstract

Quantum mechanically, an electron moving in a uniform magnetic field forms Landau levels. A curious feature is that for states with a negative angular quantum number, the total probability current vanishes, which appears to contradict the classical picture of cyclotron motion. While a geometric interpretation based on classical orbits exists, alternative interpretations remain of interest. In this paper, we examine the probability current density and identify a critical radius that naturally partitions the plane into an inner clockwise-flow region and an outer counterclockwise-flow region. We show that the vanishing total current results from an exact cancellation between these two regions. Furthermore, by defining a partitioned kinetic angular momentum with respect to the critical radius, we reveal an intrinsic competitive structure: the electron simultaneously carries two opposing rotational components. The negative quantum number manifests in the strength of the inner counter-rotation, while the net kinetic angular momentum remains positive. This bidirectional flow picture also provides a dynamical interpretation of the infinite degeneracy of Landau levels.

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9 pages, 1 figure

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Quantum mechanically, an electron moving in a uniform magnetic field forms Landau levels. A curious feature is that for states with a negative angular quantum number, the total probability current vanishes, which appears to contradict the classical picture of cyclotron motion. While a geometric interpretation based on classical orbits exists, alternative interpretations remain of interest. In this paper, we examine the probability current density and identify a critical radius that naturally partitions the plane into an inner clockwise-flow region and an outer counterclockwise-flow region. We show that the vanishing total current results from an exact cancellation between these two regions. Furthermore, by defining a partitioned kinetic angular momentum with respect to the critical radius, we reveal an intrinsic competitive structure: the electron simultaneously carries two opposing rotational components. The negative quantum number manifests in the strength of the inner counter-rotation, while the net kinetic angular momentum remains positive. This bidirectional flow picture also provides a dynamical interpretation of the infinite degeneracy of Landau levels.
We study two interacting electrons in a two-dimensional system under a strong magnetic field and show that their numerically exact solutions organize into a set of {\em sub-Landau levels} characterized by relative angular momentum quantum number mmm. These sub-levels define correlation-resolved subspaces of the Landau-level Hilbert space, while retaining the full degeneracy associated with center-of-mass motion. Within this structure, the accessible states in each correlation channel are effectively reduced, leading to a natural organization of guiding-center states consistent with a fractional occupancy. We further analyze the role of electron correlation, Zeeman splitting, and disorder in stabilizing spin-polarized electron-pair states. Building on the two-electron states, we construct a class of many-electron trial wavefunctions based on correlated electron pairs with fixed mmm, which encode short-range correlations through the vanishing of the pair wavefunction at small separation. Our results establish a direct connection between exact two-body physics and the organization of correlated many-electron states in the lowest Landau level, providing a microscopic perspective on how relative angular momentum structures can underpin the emergence of correlated phases in quantum Hall systems.
The classical problem of planar cyclotron motion of a charged particle in a uniform magnetic field possesses symmetries which account for the ``accidental'' degeneracies of the analogous nonrelativistic Schrödinger equation, as found by Johnson and Lippman. The essentially quadratic nature of the Hamiltonian is not changed by considering the particle moving in a harmonic oscillator potential, a ``Zeeman effect'' for the harmonic oscillator. The transitions to the limiting cases of a weak magnetic field (pure harmonic oscillator) or a strong field (pure cyclotron motion) involve the contraction of the corresponding symmetry groups, yielding Larmor precession of the oscillator orbits in the first case, and the drift of the cyclotron orbit in the second. The constants of the motion generate the unitary unimodular group SU2 in all cases except for pure cyclotron motion, in which case one obtains the commutation rules of creation and annihilation operators. Only for certain ratios of magnetic field strength to the oscillator frequency does one obtain bounded closed orbits, and presumably only in these cases do degeneracies exist quantum-mechanically. A transition to a rotating coordinate system reduces the problem to that of a plane harmonic oscillator; however, the time dependencies of the transformation must be allowed for interpreting the constants thereby arising. Moreover, the velocity-dependent forces introduce gauge transformations which also affect the interpretation of the symmetries. There are two kinds of symmetries—inner symmetries involving the canonical coordinates and governing the shape of the orbits, and outer symmetries involving the mechanical coordinates and governing the location of the orbits.
Abstract In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z -axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a + and a − . We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency ω c = ∣ q ∣ B m and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a + in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a − . The phase of the circular motion is the same as the phase of the complex eigen value of a + . So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z -axis. We also show that as the magnitude of the eigen value of a + becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.
We show that the newly found complete and orthonormal representation , which is constructed in terms of guiding centers and canonical momenta of an electron moving in a uniform magnetic field, also helps us to establish a systematic phase operator theory. The unitary phase operator, interpreted as a quantity canonically conjugated to the electron angular momentum, manifestly exhibits in the representation its phase behavior. As a byproduct, the phase distribution function, in the correct sense of probability in quantum mechanics, can also be defined by the representation.
We consider the orbital angular momentum of a free electron vortex moving in a uniform magnetic field. We identify three contributions to this angular momentum: the canonical orbital angular momentum associated with the vortex, the angular momentum of the cyclotron orbit of the wavefunction, and a diamagnetic angular momentum. The cyclotron and diamagnetic angular momenta are found to be separable according to the parallel axis theorem. This means that rotations can occur with respect to two or more axes simultaneously, which can be observed with superpositions of vortex states.
In this paper we demonstrate a surprising aspect of quantum mechanics that is accessible to an undergraduate student. We discuss probability backflow for an electron in a constant magnetic field. It is shown that even for a wavepacket composed entirely of states with negative angular momentum the effective angular momentum can take on positive values in some regions of space. This is also a demonstration of the azimuthal angle–angular momentum uncertainty relation. This example is chosen because the backflow motion persists indefinitely. However this work can easily be extended to other systems where the motion is notionally circular, such as the hydrogen atom.